In the crowded race for mayor of New York City, Zohran Mamdani has become a magnet for attention, ascending in the polls and raising money through a mix of social media savvy and a plain-spoken, everyman approach.

He has paid house visits to some of his thousands of small donors and taken a New Year’s Day plunge into the ocean to dramatize a rent-freeze. He broke a Ramadan fast by eating a burrito on the Q train, then faux apologized for the breach of subway etiquette — all in a breezy style more reminiscent of “Saturday Night Live” than a political ad.

But for Mr. Mamdani, a 33-year-old state assemblyman from Queens, winning the Democratic primary in June still represents a daunting challenge that goes beyond trying to convert social media virality into votes.

While many of his progressive rivals in the race have adopted more centrist positions on certain issues like policing and public safety, Mr. Mamdani, a democratic socialist, continues to embrace left-leaning views that have become less popular with voters in New York.

It has nonetheless proved to be an effective campaign strategy. Mr. Mamdani has become the standard-bearer for progressive Democrats as a fresh-faced alternative to his more veteran rivals, most notably former Gov. Andrew M. Cuomo, Mayor Eric Adams and Brad Lander, the city comptroller.

But winning an election on a hyper-progressive platform will be a challenge. Mr. Mamdani acknowledges this and knows he must get his supporters — many on the far left and outside the city’s traditional power structure — to turn out in droves.

In a primary that rarely sees turnout exceed one-fourth of eligible voters, winning over new voters could offer Mr. Mamdani outsize influence in its outcome.

At a recent campaign visit to the MAS Bronx Muslim Center in the East Bronx’s Little Yemen neighborhood, Mr. Mamdani implored those gathered to more fully use their electoral power. More than 350,000 of New York’s roughly one million Muslims are registered to vote, according to figures from the Council on American-Islamic Relations. But in the last mayoral election, only about 12 percent of them cast a ballot.

“I don’t blame anyone in our community for not voting, because oftentimes it feels like there isn’t much to vote for,” Mr. Mamdani told the group of about 100 people. “But this June 24, in this Democratic primary, we have a chance — an opportunity — to tell the world that Muslims don’t just belong in New York City but that we belong in City Hall.”

He asked congregants whether they had enough money to pay for rent, groceries, child care and their electric bills. Many sat at rapt attention, nodding their heads as he spoke. He recited the teachings of the Prophet Muhammad in arguing that their shared goal should be to make people’s lives better.

“Spread glad tidings,” Mr. Mamdani said to the group in explaining that his campaign was focused on making the city more affordable, calling it “a campaign to allow New Yorkers to dream once again.”

Mr. Mamdani underlined his campaign’s core staples: free buses, a rent freeze and city-owned grocery stores.

Campaigns typically target so-called triple prime voters who have cast ballots in three consecutive primaries. Mr. Mamdani has turned his attention to those who do not regularly hear from political campaigns. He is betting that his unrelenting focus on the cost of living will resonate with people who feel ignored by the government — a strategy he hopes will appeal to disaffected Trump voters, especially in the working-class neighborhoods outside Manhattan.

Even the conservative Manhattan Institute recognizes Mr. Mamdani’s momentum, citing his appeal to people whom the pollster John Della Volpe from the firm SocialSphere called “discontented strivers” — working-class New Yorkers who are progressive but concerned about public safety and feel that getting ahead is too difficult.

“I want you to entertain the idea that socialist Zohran Mamdani could actually become the next mayor of New York City,” the Manhattan Institute’s newsletter read. “I know it sounds crazy, but we live in strange times.”

Mr. Mamdani has already raised roughly $7 million since entering the race in October, including public matching funds. In spite of his once-limited name recognition, he now boasts more than 16,000 individual donors. Most polls show him in third place, just behind Mr. Adams and candidates with citywide political pedigrees.

“He is creating excitement within an electorate that doesn’t always see themselves reflected in leadership in New York City,” said Jasmine Gripper, a co-director of the left-leaning Working Families Party. “He is talking to a base of voters who are excited to have a candidate that holds their values, who looks like them, comes from their community, and he’s leaning in to that.”

Younger voters have been turning out with more frequency. Roughly one in five voters under 40 cast a ballot in the 2021 mayoral primary, up from about 13 percent of those voters in 2013, according to findings from the New York City Campaign Finance Board. Overall turnout, at about 27 percent of registered voters, was among the highest in recent election years.

Under the ranked-choice system, which allows voters to choose as many as five candidates in order of preference, an expansion of Mr. Mamdani’s base could help some of his more like-minded rivals. If he finishes third or worse, his votes could go to his supporters’ next-ranked candidate.

Mr. Mamdani has already committed to cross-endorsing at least one yet-unnamed opponent in an effort to stunt the momentum of Mr. Cuomo, who leads polling by a wide margin. The Working Families Party will also endorse a slate of progressive candidates and is encouraging voters to “D.R.E.A.M.” — Don’t Rank Eric or Andrew for Mayor. Several candidates, including Mr. Mamdani, support the idea.

Mr. Cuomo has already made running against the “far left” a feature of his campaign without mentioning Mr. Mamdani by name. The former governor often calls defund the police the “three dumbest words ever uttered in politics,” even though he signed some police reforms into law. Mr. Mamdani, by contrast, has called for some cuts to police spending in areas like its communications office and strategic response groups.

Trip Yang, a Democratic strategist who is not associated with any mayoral campaign, said Mr. Mamdani’s outreach to young voters in Brooklyn and Queens, and increased interest from Southeast Asian voters and Muslims, could stretch the electorate in a way unseen in recent memory.

“His policy proposals don’t seem radical,” Mr. Yang said, referring to Mr. Mamdani’s affordability platform. “The only thing radical about Zohran is probably his open democratic socialist affiliation.”

The New York City Democratic Socialists of America view that as a positive. The group has added 1,500 new members from more diverse racial and age groups since it endorsed Mr. Mamdani in October, its leaders said. They pointed to Mr. Mamdani’s heated confrontation of Tom Homan, the White House’s top immigration enforcer, at the State Capitol in Albany recently as a moment when he successfully channeled many New Yorkers’ frustrations.

“People want to fight, defend their rights and fight the authoritarian policies of the Trump administration,” said Gustavo Gordillo, the group’s co-chair. “But that’s not enough — we also have a vision for going on offense, and Zohran’s campaign has provided a vehicle for that vision.”

Some of Mr. Mamdani’s rivals have taken note of his potential, and sought to attack him where he may be vulnerable.

New York City is home to the largest Jewish population outside of Israel, and Mr. Mamdani has been criticized for accusing Israel of committing genocide in the war in Gaza. He has sponsored a bill that would prohibit New York charities from funding certain organizations that he said were tied to “Israeli war crimes.”

Whitney Tilson, a hedge fund manager and mayoral candidate, recently sent out a fund-raising email with the subject line “Stop Mamdani” citing his “far-left platform,” “fiery rhetoric against N.Y.P.D. and Israel” and support from professors at Columbia University, where his father is a professor. The Trump administration has accused the university of not doing enough to quell antisemitism on campus.

But Mr. Mamdani’s opponents also have their own vulnerabilities to deal with.

At some of the mosques where Mr. Mamdani has appeared, members have approached him with stories, and occasionally pictures, of Mr. Adams speaking to them. They recalled how the mayor spoke about growing up in a working-class household and pledged that his administration would be about “delivering the dignity they’d been denied” at City Hall.

“The reason that we’re in this moment is that he betrayed those voters,” Mr. Mamdani said of the mayor. “We’re trying to keep our promise.”