To the crowd of voters who gathered in the garden of a New Hampshire home on a serene summer’s day in June, Elizabeth Warren expressed her sincere gratitude.

There was a dissonance, she acknowledged, between the “gorgeous” setting and the bleak picture she would paint of a democracy in peril.

“Our country’s in real trouble,” Warren said, pacing as she spoke. “A lot of trouble.

“So what’s it going to take to change that?” she continued. “Big, structural change.”

Alexandra Barker, the host of the event, couldn’t agree more. “If things don’t change, this is the home I’ll sell to send him to college,” she said, hugging her 11-year-old son, Ricky. “If there’s anyone who can actually get it done, I believe it’s going to be her.”

More than six months into her presidential campaign – after holding 105 town halls, answering nearly 500 questions from voters and taking more than 35,000 selfies – the Massachusetts senator has emerged as a top-tier contender, and increasingly as the top choice of the American left, in the crowded race for the Democratic nomination.

Elizabeth Warren at a campaign rally in Fairfax, Virginia, on 16 May. Photograph: Cliff Owen/AP

But the opening months of Warren’s campaign were less even. She was the first major candidate to enter the race on New Year’s Eve 2018 amid a cloud of controversy over her past claims of Cherokee heritage and speculation that she would run in the shadow of her longtime ideological ally and closest rival, Bernie Sanders.

Her decision to release the results of a DNA test that revealed her distant Native American ancestry in response to taunts by Donald Trump upset members of the Cherokee nation and continues to raise questions about her political acumen among some voters desperately trying to unseat the president.

Setting the agenda

Warren’s ascent has been fueled in part by her rapidly expanding portfolio of liberal policy proposals for establishing universal childcare, canceling student loan debt, breaking up big tech and expanding voting access.

Her centerpiece proposal – the plan to pay for her plans – is an “ultra-millionaire tax” on assets above $50m, which she tells audiences would include “your Rembrandts, your stock portfolio, your diamonds and your yachts”. Taken together, her platform would dramatically reshape the American economy.

The steady stream of policy releases has boosted Warren’s profile and, importantly, created fundraising opportunities for a campaign that has staked its financial sustainability on grassroots support. By offering a clear and detailed vision of what a Warren presidency might look like, she has forced her rivals to play catch-up.

Her standing was cemented late last month by a crisp performance in the first presidential primary debate in Miami. A spate of recent post-debate polls showed the former Harvard law professor had gained ground, alongside the California senator Kamala Harris, as support for the early frontrunners Joe Biden and Sanders waned. She has ranked consistently, placing third in three national surveys and one poll of Iowa caucusgoers.

Elizabeth Warren speaks with Julián Castro and Cory Booker at the first Democratic debate in Miami, Florida, on 26 June. Photograph: Mike Segar/Reuters

“She has the professor part down,” said Lucy Gagnon, after waiting patiently in one of Warren’s famous “selfie lines” for her chance to take a photo with the senator at an event in Windham.

“I really believe that she could educate the country as to what these issues really are and to make them understandable for people because I don’t think the average person realizes what trouble we’re in.”

Maria Velez, who came to Warren’s town hall on the Florida International University campus in Miami the day before, said she had never donated to a candidate before sending $20 to Warren’s campaign – twice.

“Every time I hear her talk I’m inspired,” Velez said. “She has a plan for everything. And she explains her plans in a way that is so easy to understand.”

Every time I hear her talk I’m inspired. She has a plan for everything. And she explains her plans in a way that is so easy to understand Maria Velez

Warren, like others in the race, has faced skepticism about her “electability” – a concern that can include gender bias – as a party still carrying the scars of Hillary Clinton’s 2016 loss tries to divine which candidate is best suited to defeating Trump.

David Darnell, who saw Warren speak at the house party in New Hampshire, said he likes Warren’s policies but was initially put off by her “finger-wagging, schoolmarm” presentation on the campaign trail.

“I want to stress, one, I think she’s gotten better,” Darnell said. “But, two, I think it’s a problem we white males have, we don’t like it when someone speaks down to you. And I hate to admit [that] it bugged me.”

Warren is among his top choices, alongside Senators Harris, Cory Booker and Amy Klobuchar. But he worries Warren, who Trump has mocked as “Pocahontas” might be too easy a target for the president.

When asked about her “electability”, Warren points to her 2012 Senate race against Scott Brown, a popular Republican incumbent. The race was fiercely contested and polling at the time found that Brown had a higher favorability.

Warren defeated Brown to became the first female senator in Massachusetts’ history.

‘My story has a lot of twists and turns’

It’s not just her plans that are resonating. In interviews with more than a dozen attendees at Warren’s campaign events in recent weeks, people, and especially women, said they were drawn to her personal story.

“I’m like a lot of Americans,” Warren says. “I do not have an entirely straight path story. My story has a lot of twists and turns.”

Warren grew up in Oklahoma, the youngest of four children. In the early 1960s, her father suffered a heart attack and lost his job. The family lost their station wagon and they would have lost their house, too, Warren says, if her mother, who had never worked outside the home, hadn’t found a minimum-wage job at the local Sears department store.

“When I was a girl, a full-time minimum-wage job in America would support a family of three,” Warren said at the house party in New Hampshire.

“Today a full-time minimum-wage job in America will not keep a momma and a baby out of poverty. That is wrong, and that is why I’m in this fight.”

She unspools the story of her early adulthood emphatically, bit by bit. She won a scholarship to college (“Yay!”) , then she fell in love at 19, got married (“Not to that guy,” she says, pointing theatrically at her second husband, Bruce Mann); and dropped out of college (“Uggh.”)

She later enrolled at a commuter college and eventually landed her “dream job” as a special needs teacher. But after becoming pregnant, she was “shown the door”.

Shortly after graduating from law school, a feat she credits to the M&Ms that helped potty-train her children so she could leave them at daycare while attending class, Warren took a university job teaching law. She married Mann, whom she calls her “keeper husband”, and set about becoming one of the nation’s leading experts on bankruptcy law.

Before answering questions from the audience, Warren typically wraps her story: “My daddy ended up as a janitor but his baby daughter got the opportunity to become a public school teacher, to become a college professor, to become a United States senator and to become a candidate for president of the United States.”

‘The biggest anti-corruption plan since Watergate’

Warren is gaining ground with her plans and her story. But what Democratic voters say they care about most is a plan to remove Trump from office.

Warren believes the answer is to take Trump at his word. In 2016 he promised to “drain the swamp” and fix Washington. But she argues that Trump has only made Washington swampier.

Corruption, she believes, is the root of Washington’s dysfunction and the reason Congress is not responsive to popular policy ideas. The next president must first address corruption if Democrats hope to make progress toward any of their policy goals, from enacting immigration reform to protecting the planet.

“Here’s the good news,” she likes to say. “I have the biggest anti-corruption plan since Watergate. Here’s the bad news. We need the biggest anti-corruption activist since Watergate.”

To emphasize her commitment to fighting corruption, Warren swore off private fundraising events with wealthy donors during the primary race. It’s a decision that could later put her at a financial disadvantage against candidates like Biden, who has relied on traditional high-dollar fundraisers, as well as Harris and South Bend’s mayor, Pete Buttigieg, who have also attracted big donors.

Elizabeth Warren walks past the Homestead detention center, where migrant teens are held, in Homestead, Florida, on 26 June. Photograph: Daniel A Varela/AP

Warren’s campaign says it isn’t worried. Forgoing high-dollar fundraisers gives her more time to spend hosting town halls and posing for photos with voters, aides say. And her slow and steady, one-selfie-at-a-time approach to the race has worked so far.

But before taking on Trump, she must contend with the shifting dynamics of the primary race, as she chases Sanders on her left and fends off rivals vying to be the alternative to Biden, who remains the early frontrunner.

She also faces long-term challenges expanding her base to include more people of color and moderate voters, key Democratic constituencies. Polling shows she has made gains among black voters, a key constituency for Democrats. On Friday she introduced a series of executive actions she would take to level the wage and leadership gap for women of color.

There are also fresh signs that a Warren nomination could face less institutional resistance should she emerge as the party’s nominee. At a recent gathering of moderate Democrats and lawmakers, several acknowledged that they had policy quibbles with Warren but otherwise weren’t opposed to her nomination.

But most urgently, she must move past Sanders, whose strong grassroots support and loyal base represents a unique challenge.

With her ferocious critique of 21st century capitalism – as well as her quick call for Trump’s impeachment, her refusal to go on Fox News and her resolute support for eliminating private health insurance, a position that some activists had started to doubt – Warren has steadily carved into the progressive coalition Sanders built in 2016.

At the same time they have distanced themselves from one another. Sanders proudly defines himself as a democratic socialist while Warren calls herself a “capitalist to my bones”.

And therein lies her grandest plan: to save capitalism from itself.

“When you’ve got a government – when you’ve got an economy that does great for those with money and isn’t doing great for everyone else – that is corruption, pure and simple,” she said from the center of the debate stage in Miami earlier this month. “We need to call it out. We need to attack it head on and we need to make structural change in our government, in our economy and in our country.”