WASHINGTON — For well over a year now, President Biden’s vaunted negotiating style largely boiled down to this: I’m with you.

After he vanquished Senator Bernie Sanders of Vermont in the Democratic primary, he brought the liberal icon’s ardent supporters into the fold by embracing much of the senator’s platform even as he ran on unifying the country. When moderate Democrats came to call, he used the tones of centrism to assure them of his conciliatory bona fides.

But when Mr. Biden ventured to the Capitol on Friday to help House Democrats out of their thicket, he had to choose sides. He effectively chose the left.

“The way he is governing doesn’t reflect the skills I know he must have from his years as a legislator,” said Representative Stephanie Murphy of Florida, who had been one of the moderate Democrats demanding an immediate vote on a trillion-dollar infrastructure bill, convinced that was what the president wanted — or at least needed. She called Mr. Biden’s refusal to push harder for legislation he had embraced “disappointing and frustrating.”

“I’m not clear why he came up to the Hill,” she grumbled.

Since the president claimed his party’s nomination last year, he has nurtured the fragile peace between his party’s fractious center and left by convincing both sides he is their ally. Unified first by their shared disdain for former President Donald J. Trump, and then by Mr. Biden’s adoption of an expansive platform, the two factions remained in harmony into this year. They responded to the pandemic by passing a sweeping stimulus package in the spring.

Now, the two factions are at loggerheads — one flexing its power but as yet is empty-handed, the other feeling betrayed, both claiming they have the president on their side — and the outcome of their battle over Mr. Biden’s proposals could determine Democrats’ fate in the midterms and the success of his presidency.

That agenda consists of two sweeping domestic proposals resembling a modern Great Society: the “American Jobs Plan,” spending $1 trillion over 10 years on traditional infrastructure like roads, bridges and tunnels, and a bigger and more controversial “American Family Plan,” which the Democrats labeled “soft infrastructure” — including universal prekindergarten and community college, paid family and medical leave, child care and elder care support, and an expansion of Medicare.

But liberals feared that moderate Democrats would vote for the infrastructure bill, claim victory, and peel away from the social policy measure, so they refused to support the smaller infrastructure bill until the larger social-policy package had been passed.

Heading into last week, both the moderates and the progressives felt as if they had ironclad promises: the moderates, that a vote on infrastructure would happen before October; the liberals, that the bill, a crucial part of the president’s domestic agenda, was inextricably twinned with their higher priority, the more expansive measure addressing climate change and the frayed social safety net.

The liberals, however, used their larger numbers to blockade the infrastructure bill — and they said they did it for Mr. Biden. Representative Ilhan Omar, a left-wing Democrat from Minnesota and one of the leaders of the blockade, stood before reporters last week and said the blockaders were the ones “trying to make sure that the president has a success.”

“If we pass the infrastructure bill alone, we are not even accomplishing 10 percent of his agenda,” said Ms. Omar, the vote-counter in the Congressional Progressive Caucus, a bloc of Democrats nearly 100-strong, who showed their cohesion in last week’s showdown.

This enraged both the nine centrist lawmakers who had forced Speaker Nancy Pelosi to promise an infrastructure vote by the end of September, and a larger, quieter group of backbench House Democrats, many from swing districts, who were eager for the president to sign the public works bill and start trumpeting the funding for roads, bridges and broadband in their districts, at a time when Mr. Biden’s approval ratings were sagging.

“I don’t think it’s good for the Joe Biden administration, and I don’t think it’s good for Democrats,” said Representative Henry Cuellar of Texas, a Democrat, suggesting that Mr. Biden was effectively siding with the left by not lobbying for passage of the infrastructure package.

In part, that anger stemmed from Mr. Biden’s go-along-to-get-along style.

“You got the feeling that Uncle Joe is for everybody, he likes everybody,” said Representative Emanuel Cleaver of Missouri.

Members of the moderate wing were explicit on Friday, blaming the liberals but also insisting that they themselves were Mr. Biden’s true torch bearers. Representative Josh Gottheimer, Democrat of New Jersey, denounced a “small faction on the far left” that he said had employed “Freedom Caucus tactics” to “destroy the president’s agenda” — a reference to the hard-right faction of the House that bedeviled Republican leaders when they were in charge.

“We were elected to achieve reasonable, common-sense solutions for the American people — not to obstruct from the far wings” Mr. Gottheimer fumed in a statement released late on Friday night. “This far-left faction is willing to put the president’s entire agenda, including this historic bipartisan infrastructure package, at risk. They’ve put civility and bipartisan governing at risk.”

Given the range of the party’s suburbanites-to-socialists coalition, it may have been inevitable that Mr. Biden would eventually anger one wing of his party. What was striking, and perhaps equally surprising to both blocs, was that he alienated the moderates who had propelled him to the nomination while delighting the progressives who vociferously opposed him in the primary.

The president is not backing off the public works measure so treasured by the moderates.

But as he told House Democrats on Friday, he believes it’s “just reality” that the infrastructure legislation will not pass without assurances from the centrist Senators Joe Manchin III of West Virginia and Kyrsten Sinema of Arizona that they will support the more wide-ranging bill.

Though, as Mr. Biden conceded in the Capitol, that won’t happen until the more expansive bill is pared back to meet the two senators’ approval.

The decision to keep the fate of each bill tied to the other’s measure amounts to a gamble. Infrastructure was the bird in hand; it passed the Senate with bipartisan bonhomie in August with 69 votes.

Together, they are in trouble, which deepens with every new demand by Mr. Manchin and Ms. Sinema that pulls the social policy bill further from the liberals’ vision. If the two factions cannot agree on that measure, Mr. Biden might end up with nothing — a catastrophic blow for his party and its leader.

Delaying the infrastructure bill is not, as Representative Dean Phillips of Minnesota put it, “the linear and expeditious path to which most of us would aspire.”

Mr. Phillips, a well-liked moderate who captured a Republican district in 2018, expressed hope earlier in the week that Mr. Biden could serve as a bridge between the party’s factions. But he acknowledged on Friday that those chances had “been sadly diminished” in light of what he called the president’s “nothing-burger” of a visit to the Capitol.

Mr. Phillips said he thought both bills would still get done. But, privately, other lawmakers from competitive seats were disconsolate that they would not be able to spend the remainder of this fall holding up evidence of bipartisan achievement in Washington.

Mr. Biden is eager to sign both bills. One of his aides on Friday likened them to children he loves equally.

That has not, though, stopped both factions of the party from claiming that they are the ones seeking to assure passage of his agenda.

The result is quite a turnabout.

“We are fighting for the Build Back Better agenda,” said Ms. Omar, employing Mr. Biden’s preferred slogan — which would have been shocking at this time two years ago, when she rallied early to Mr. Sanders’s candidacy.

Throughout 2019 and in the first months of 2020, Mr. Biden was an object of scorn from the left. He was too old, too moderate and an obviously bad fit for an increasingly young, diverse and progressive party, they said, often mocking him in harsh terms.

Mr. Biden believed liberals were the ones out of step with the Democratic center of gravity. And he effectively proved it by assembling a multiracial coalition that was animated by defeating Mr. Trump more than by any bold policy agenda.

Yet because his primary had largely centered on ousting Mr. Trump and unifying the country, he had little in the way of firm policy plans. And in making peace with progressives after he secured the nomination, he adopted a number of their ideas.

That has allowed left-wing Democrats to say, with wide smiles, that they are only trying to fulfill Mr. Biden’s vision. The question now is whether his attempt to pass both bills will pay off — or if his decision to not push for quick passage of the infrastructure bill will leave him with a protracted standoff, or nothing at all.

What’s certain, however, is that after Mr. Biden’s all-things-to-all-people campaign, he has committed himself to many of the policies that his liberal critics were skeptical he would embrace.

“For all of the progressives who kept telling me there was no difference between Joe Biden and Mike Bloomberg,” said Representative Brendan Boyle, an early Biden supporter from Philadelphia, “where Biden has come down in this internal debate shows how absurd that claim always was.”