In the first exchange, Ms. Warren, who is from Massachusetts, was asked repeatedly how she would pay for “Medicare for All” and insisted that costs would not increase for middle-class Americans. But she would not say outright that taxes would or would not increase.
Bernie Sanders, the Vermont senator who has championed a single-payer system, said taxes would go up for “virtually everybody.” But he added that the tax increase would be substantially less than what people currently pay for premiums and out-of-pocket health care expenses.
“At least Bernie is being honest here, and saying how he’s going to pay for this and that taxes are going to go up,” Ms. Klobuchar, of Minnesota, said after his comments.
“And I’m sorry, Elizabeth, but you have not said that and I think we owe the American people to tell them where we will send the invoice.”
Ms. Warren also came under attack over health care from Pete Buttigieg, the mayor of South Bend, Ind., who criticized her for being evasive.
A short while later, Ms. Klobuchar again addressed Ms. Warren during an exchange about a wealth tax.
“I want to give a reality check to Elizabeth,” she said. “No one on this stage wants to protect billionaires. Not even the billionaire wants to protect billionaires,” she added, in a reference to Tom Steyer.
“We just have different approaches. Your idea is not the only idea.”
In response, Ms. Warren reiterated her belief that a wealth tax was necessary in addition to income taxes.
“I think as Democrats we are going to succeed when we dream big and fight hard, not when we dream small and quit before we get started,” Ms. Warren said.
Now in her third term, Ms. Klobuchar was the first woman elected to the Senate from Minnesota, and she remains hugely popular there. She has focused on issues like prescription drug costs, sexual harassment and online privacy. A 2016 analysis found that she had passed the most laws of anyone in the Senate.
She announced her campaign on a snow-covered stage in Minneapolis in February and vowed to rejoin the Paris climate agreement on her first day in office. Her team hopes that her brand of “Minnesota nice” politics will attract the Iowa voters who cast the first primary votes, and residents in states like Wisconsin, Michigan and Pennsylvania that swung the 2016 election to President Trump.
But she has lagged behind other Democrats in the polls, with a national polling average of 2 percent before the debate started on Tuesday. She had raised $9.1 million in individual donations through June 30. Senator Bernie Sanders of Vermont, the leader in fund-raising in that period, had raised $36.2 million.
Ms. Klobuchar has also been known as a demanding boss. In February, The Times reported that some former aides said that she was often dehumanizing. (They recounted a particularly strange 2008 episode involving a salad and a comb.) In response, Ms. Klobuchar conceded that she pushed her staff hard, but said some of the anonymous stories circulating were “just plain ridiculous.”
A former corporate lawyer and prosecutor, Ms. Klobuchar also gained national attention during the Senate Judiciary Committee hearing for Justice Brett M. Kavanaugh, when she pressed the nominee on whether he had blacked out drinking.
“Have you?” he shot back, to which she calmly replied, “I have no drinking problem, Judge.”
The exchange went viral, in part because Ms. Klobuchar mentioned that her father battled alcoholism, and led to a parody on “Saturday Night Live.”