Democrats and Republicans are already jockeying for a crucial voting bloc that soured on Donald Trump, tilted to Joe Biden and now holds the key to the second half of the president’s term.PAPILLION, Neb. — Pursuing a bipartisan infrastructure deal and trumpeting a revived economy and progress against the pandemic, President Biden is trying to persuade the nation that Democrats are the party that gets things done. His message is aimed at holding on to a set of voters in next year’s midterms who could determine the fate of his agenda: suburbanites who abandoned former President Donald Trump in droves.More than any other group, those independent-minded voters put Mr. Biden in the White House. Whether they remain in the Democratic coalition is the most urgent question facing the party as it tries to keep its razor-thin advantage in the House and the Senate next year. Mr. Biden made his pitch again on Friday when he signed an executive order intended to protect consumers from the anti-competitive practices of large businesses. But Republicans are also going to war for suburban votes. The party is painting the six-month-old Biden administration as a failure, one that has lost control of the Southwestern border, is presiding over soaring crime rates and rising prices and is on the wrong side of a culture clash over how schools teach the history of racism in America.Whoever wins this messaging battle will have the power to determine the outcome of the rest of Mr. Biden’s term, setting the stage for either two more years of Democrats driving their policies forward or a new period of gridlock in a divided Washington.Both parties are targeting voters like Jay Jackson, a retired career Air Force officer who is now a reservist in the Omaha suburbs. Mr. Jackson had lawn signs last year for Republicans running for Congress, but also for Mr. Biden. He thought that Mr. Trump had failed to empathize with military duty and regularly lied to Americans, and did not deserve re-election.“I’m a classic RINO,” Mr. Jackson said with a laugh, accepting the right’s favorite insult for voters like him: Republicans in Name Only. In a guest column in The Omaha World-Herald, Mr. Jackson, a 39-year-old lawyer, explained his view: “We Republicans need to turn away from Trump and back to our values and the principles of patriotism and conservatism.”Mr. Biden won 54 percent of voters from the country’s suburbs last year, a significant improvement over Hillary Clinton in 2016, and enough to overcome Mr. Trump’s expansion of his own margins in rural and urban areas, according to new data from the Pew Research Center. Suburbanites made up 55 percent of the Biden coalition, compared with 48 percent of Clinton voters.Jay Jackson encouraged fellow Republican voters to “turn away from Trump.”Walker Pickering for The New York TimesLia Post voted routinely for Republicans but supported Mr. Biden last year.Walker Pickering for The New York TimesThe authoritative Pew study, which echoed other recent surveys, also showed that Mr. Biden failed to increase his share of the Democratic base from 2016, including among young people and voters of color. It found, however, that his support surged among independents, veterans and married men — voters like Mr. Jackson.But even as Mr. Jackson crossed party lines for Mr. Biden, he supported Representative Don Bacon, a Republican who won re-election in Nebraska’s Second District, which Mr. Biden himself carried. Mr. Jackson said that he was pleased so far with the Biden administration — especially its “putting the accelerator to the floor on Covid” — but that he would very likely vote again for Mr. Bacon.It shows that in 2022, Democrats will need to count on more than the revolt of suburbia against Mr. Trump’s norm-smashing presidency to motivate their voters.The limits of the anti-Trump vote were already glimpsed last year, when half of the 14 House seats that Democrats lost, to their shock, were in suburban or exurban districts. The party also failed to defeat vulnerable Republicans in districts Mr. Biden won, such as Nebraska’s Second.For 2022, Democrats’ congressional finance committee has identified 24 “frontline” incumbents in swing districts, some two-thirds of them in suburban areas.Representative Sean Patrick Maloney of New York, the chair of the Democrats’ election arm, aims to fuse Republican candidates with Mr. Trump’s divisiveness and with the party’s obstruction of gun restrictions, expanding health care access and fighting climate change.“The post-Trump Republican brand is bad politics in the suburbs,” he said in an interview. “They have embraced dangerous conspiracy theories, flat-out white supremacists and a level of harshness and ugliness that is not appealing to suburban voters.”Representative Tom Emmer of Minnesota, who leads the G.O.P. campaign arm, said Republicans would attack Democrats over a set of “incredibly toxic” issues for the suburbs. He listed them as crime, tax increases, border security and the latest flash point of the culture wars, critical race theory — the idea that racism is woven into American institutions, which Republicans have seized on in suburban school districts.Sarpy County is the fastest-growing county in Nebraska, with young newcomers drawn to jobs in tech or in Omaha’s insurance industry, and to the exploding housing market.Walker Pickering for The New York Times“It’s going to be a big issue in 2022,” Mr. Emmer said.He added that while Democrats “seem to be focused on a personality in the past” — Mr. Trump — “we’re focused on issues.”House Democrats also face structural and historical obstacles to retaining their slender nine-seat majority. In the modern era, a president’s party has lost an average of 26 House seats in midterm elections. Redistricting will place nearly all members of the chamber in redrawn seats, with Republicans wielding more power to gerrymander than Democrats.National polling shows Mr. Biden’s job approval consistently above 50 percent. But some recent surveys of swing House districts suggest that the president is less popular on specific issues. A survey in May of 37 competitive House districts by a Democratic group, Future Majority, found that more voters disliked than liked Mr. Biden’s handling of the economy, climate policy and foreign affairs. He was especially unpopular over the U.S.-Mexico border and relations with China.But Val Arkoosh, a Democratic official in the Philadelphia suburbs who is running for the Senate in 2022, said that issues that rally Democrats, like voting rights and health care, would still be on the ballot, even if Mr. Trump — who drove furious opponents to the polls last year — is not. “Yes, the former occupant of the White House is gone, but we continue to see a significant amount of obstruction in Washington around issues people here care deeply about,” she said.While suburbs across the country vary demographically and politically, the independent voters of suburban Omaha present a snapshot of the terrain where both parties will be fighting their hardest.Nebraska is one of just two states to award a share of its electoral votes by congressional district. Mr. Biden’s success in carrying the Second District, which includes Omaha and much of its suburbs, went beyond the single electoral vote he picked up. He flipped the district by 8.75 percentage points after Mr. Trump had won it in 2016 — a larger swing than in any individual battleground state.The suburban part of the district is mostly in western Sarpy County south of Omaha. It is the fastest-growing county in Nebraska, with young newcomers drawn to jobs in tech or in Omaha’s insurance industry, and to the exploding housing market.Corbin Delgado, the secretary of his party’s state Latinx Caucus, said his top issue was immigration reform.Walker Pickering for The New York TimesJen Day won a State Senate race as a Democrat, though many of her voters supported a Republican candidate for Congress.Walker Pickering for The New York TimesFields of corn race up hillsides and yield suddenly to home developments with names like the Mansions at Granite Falls. A vast Amazon distribution center that will employ 1,000 workers is under construction. A sign at another building site promises the “Future Home of Lamb of God Lutheran Church.”Older towns in the county command hilltops, their water towers visible from afar like medieval castles.Last year, Sarpy County, like most places, had higher turnout by both parties and independents compared with 2016. But the surge especially among independents probably accounts for Mr. Biden’s winning 13,000 more votes in the county than Mrs. Clinton did. (Mr. Trump’s votes increased by only about 7,000.)“We have a lot of younger families moving in,” said Charlene Ligon, an Air Force retiree who leads the county Democrats. “They may be conservative, but they’re more centrist, with younger attitudes.”Jen Day, a small-business owner in her 30s, won a State Senate race as a Democrat in November, the first time in memory the party had captured a seat in western Sarpy County.Ms. Day said many of her supporters had also voted for Mr. Bacon, the Republican congressman. “From discussions I’ve had with people in the district, I don’t think they’re pledging allegiance to either party at this point,” she said.Jeff Slobotski, a suburban father of five who changed his registration from Republican to independent, said the Bacon seat was “absolutely winnable” for Democrats in 2022. A Trump supporter in 2016, Mr. Slobotski voted for Mr. Biden last year.Mr. Slobotski, 43, is an executive for a company that brings tech start-ups and arts groups to an emerging neighborhood in the city. He spoke over lunch last week at a downtown Omaha restaurant, the Kitchen Table. The restaurant windows displayed posters for Black Lives Matter and for a young state senator, Tony Vargas, who has been mentioned as a possible Democratic nominee to take on Mr. Bacon.Fields of corn race up hillsides and yield suddenly to home developments.Walker Pickering for The New York TimesAlthough Mr. Slobotski voted for Mr. Bacon, he said he would support Mr. Vargas if he ran for the seat. “He’s just a young visionary, somebody with leadership ability, more of a pragmatist,” he said of Mr. Vargas, a former Omaha school board member. The Democrats’ 2020 nominee, Kara Eastman, was considered by many to be too progressive for the District.Later that day, at a restaurant in Papillion, a group of three other 2020 ticket-splitting voters sipped iced coffees as they assessed Washington under unified Democratic control.All three had voted for Mr. Biden, but none supported the drive by many congressional Democrats to blow up the filibuster to pass Mr. Biden’s most ambitious agenda items.These voters preferred a scaled-back infrastructure package that, even if it left major spending on education and climate on the table, could pass with bipartisan support and represent a show of unity. “It’s one of those things that kind of builds relationships to get things going,” said Michael Stark, 30, an independent.The filibuster is “there for a purpose and I am terrified of what would happen if it went away,” said Corbin Delgado, 26, a Democrat who works for a nonprofit group and is the secretary of his party’s state Latinx Caucus. He said his top issue was immigration reform, including a pathway to citizenship for the undocumented. He voted for Mr. Bacon last year, he said, because the Republican had modified his opposition to some immigration changes after meeting with activists. “I’m a big believer that when a politician actually listens and changes, that should be rewarded,” he said.But he would leap at the chance to vote in 2022 for Mr. Vargas, who represents a district with a large Hispanic population.Lia Post, 54, grew up in a conservative religious family and voted routinely for Republicans. An activist for legalizing medical marijuana, she supported Mr. Biden last year. She said that more than anything else, she was relieved by the absence of perpetual chaos in Washington.“I don’t feel so stressed out all the time,” she said. “I just feel now I have a president that I can just breathe,” she added, and not worry, “‘Oh, God, what’s the next thing?’” More