At the same time, she is spending money quickly to build up a large campaign operation. Her campaign said it spent $5.2 million in the first quarter, leaving it with $11.2 million in cash on hand.

[Check out our tracker of the 2020 Democratic candidate field.]

Ms. Warren’s fund-raising at the beginning of her campaign did not impress. On Dec. 31, the day she announced her exploratory committee for president, she raised about $300,000 through the online donation-processing platform ActBlue, according to Federal Election Commission records. By contrast, Mr. O’Rourke raised $6.1 million in his first 24 hours in the race, and Mr. Sanders raised $5.9 million in his first day, according to their campaigns.

A New York Times article detailing Ms. Warren’s early fund-raising struggles, which was published online on the last day of the first quarter, helped create an outpouring for Ms. Warren, handing her one of her best fund-raising days of the quarter, according to a Democrat familiar with her fund-raising.

Supporters of Ms. Warren feared that she was being crowded out by Mr. Sanders on the party’s left flank, but the campaign’s surge in the final week has at least temporarily allayed some of the concern. During that time, Ms. Warren’s campaign team made several dramatic pitches to its donor base, sending emails that addressed the harshest criticisms of the campaign, such as one titled “Can she really win?” in which they laid out the campaign’s overarching strategy.

The emails, which track with private conversations among Ms. Warren’s senior advisers, suggest that the senator believes several things will help her break out of the Democratic pack, including an ability to do more political events because she is not devoting time to private fund-raisers, her well-praised policy rollouts and her campaign team’s focus on digital organizing.

“The relative numbers don’t matter to me,” said John Walsh, a former chairman of the Massachusetts Democratic Party. “Who has the most money? No, that doesn’t matter. It’s, ‘Do you have enough money to run the campaign you said you’re going to run?’”

Candidates must report their fund-raising for the first quarter to the Federal Election Commission by April 15, but campaigns can announce numbers before that deadline.