The House Republican leader, Kevin McCarthy, came within striking distance of winning the speakership, in a clear sign of momentum despite losing a 13th consecutive vote on the fourth day of a grinding intra-party showdown.

Fifteen far-right holdouts dropped their opposition and voted for McCarthy, after the embattled Republican appeared willing to accept a proposal that would undermine his own power while giving the far-right flank more influence over the legislative process, including the ability to more easily remove a speaker.

“We’re at a turning point,” congressman Scott Perry of Pennsylvania, a leader of the conservative rebellion who voted for McCarthy on Friday, said. Encouraged by the contours of the emerging deal, Perry said he cast his vote in a “good-faith effort” to end the stalemate.

For the first time, McCarthy was the top vote winner, surpassing the Democrats’ choice for speaker, congressman Hakeem Jeffries of New York, though it was still not enough to cement the top job. And despite the movement, it was not clear the the proposal would placate the remaining detractors, who remained committed to denying McCarthy the 218 votes traditionally needed to secure the gavel.

“You only earn the position of speaker if you get the votes,” said congressman Matt Gaetz, a holdout who voted for conservative congressman Jim Jordan on the 13th ballot. McCarthy, he predicted, “will not have the votes tomorrow and he will not have the votes next week, next month, next year”.

Building frustration between the holdouts and the rest of the Republican conference spilled onto the chamber floor. As Gaetz assailed McCarthy in a nominating speech for his alternative speaker candidate, congressman Mike Bost, a Republican of Illinois, angrily interjected. shaking a finger at Gaetz he shouted: “This is not going to bring anyone.”

The house clerk banged the gavel, ending the exchange, but several Republicans stormed off the floor in protest as Gaetz finished his remarks. But the general mood had shifted. Republicans burst into applause with each defection from the rebel camp. McCarthy, who had watched with a forlorn expression as the spectacle of defeat without progress unfolded, was notably more upbeat, smiling and laughing as he listened to the roll call.

Until a speaker is chosen, the House will continue to operate in a suspended state of paralysis, with members unable to be sworn in and business unable to proceed.

McCarthy’s allies worked late into the night on Thursday to hammer out the details of a deal, after a third day of balloting yielded the longest succession of failed speaker votes since the Congress of 1859, which went 44 rounds and two months. Only four other speakership elections in American history have required more than 12 ballots.

“Apparently, I like to make history,” McCarthy told reporters as he left the House floor on Thursday night, following an 11th vote that produced effectively the same failed result.

A handful of members were absent for the fourth day of the speakership election, which was not originally supposed to be a voting day for the House. Congressman Wesley Hunt, a Republican of Texas, had to rush home to be with his wife and newly born son. Congressman David Trone, a Democrat of Maryland, missed the 12th vote because he was undergoing surgery. Trone returned to the Capitol for the 13th vote, receiving a standing ovation from his colleagues as he cast a ballot for Jeffries with his arm in a sling.

The House reconvened for a fourth day of voting, in the shadow of the second anniversary of the deadly Capitol assault. The shocking scenes of violence and chaos that erupted on 6 January 2021 began with an attempt by ultra-conservative lawmakers loyal to Donald Trump to stop Congress from certifying the results of the 2020 presidential election that Joe Biden won. Several of those election-denying lawmakers are leading the revolt against McCarthy.

After initially condemning the insurrection and Trump’s role fomenting it, McCarthy traveled to Mar-a-Lago in early 2021 to repair relations with the former president and his loyalists in anticipation of his speaker bid. Trump emphatically endorsed McCarthy this week, but it has so far failed to change any minds.

Before voting began, members of Congress, mostly Democratic lawmakers, gathered with the families of officers who died in the attack on the steps of the Capitol to mark the “solemn day”.

“We stand here today with our democracy intact because of those officers,” said Jeffries, the new Democratic leader. “Violent insurrectionists stormed the Capitol and attempted to halt the peaceful transfer of power, a cornerstone of our republic. They failed.”

Meanwhile, House Republicans were holding a conference call to discuss the terms of a possible deal hammered out between emissaries of McCarthy and leaders of the rightwing rebels.

“The entire conference is going to have to learn how to work together,” McCarthy told reporters before leaving the Capitol on Thursday night. “So it’s better that we go through this process right now.

To win the bloc of rebels thwarting his rise, McCarthy was apparently prepared to agree to conditions that he had not been previously willing to accept. That includes reinstating a rule that would allow a single lawmaker to force a vote to remove the speaker, effectively placing himself at the mercy of his detractors who could trigger a vote at any point.

There is a risk that Republican leadership’s myriad concessions to the party’s hard-right faction could repel more moderate members, who have so far remained loyal to McCarthy. But so far all the momentum has only trended in McCarthy’s favor.

In yet another sign of the shifting tide, the conservative grassroots group FreedomWorks, which had previously mocked McCarthy’s speakership bid, dropped its opposition to his candidacy.

“While details continue to be deliberated on McCarthy’s campaign to 218 votes,” the group’s president said, “today represents a step in the right direction to changing the way business is conducted in Congress.”

Democrats remained united behind Jeffries as their choice for speaker. He repeatedly won the most votes during the first three days of balloting, but remained short of the majority.

The spectacle foreshadows the difficulties that lie ahead for the Republican party as it aims to reclaim the Senate and the White House in 2024. Already riven by infighting, and constrained by a narrow majority, the party’s new leaders will face many of the same challenges as past Republican speakers, whose tenures were defined by government shutdowns and political brinkmanship.