“Do I think that there is room for growth,” he said, “and do I think that Senator Sanders would have liked the numbers to have been even further up among voters of color, among young voters, among working-class voters? Absolutely.”

Mr. Khanna expressed confidence that the numbers would increase in a general election contest against Mr. Trump, but said that the campaign had to “keep pushing harder.”

On the campaign trail, Mr. Sanders, 78, has proclaimed that his “is the campaign of energy, is the campaign of excitement, is the campaign that can bring millions of people into the political process who normally do not vote.” In rallies in Texas over the weekend, as his resounding victory in Nevada was becoming clear, he conveyed an air of triumph, drawing enormous crowds as his campaign made plans to solidify his front-runner status by Super Tuesday on March 3.

“If the cameras turn on this crowd, and our friends in Wall Street and the drug companies see this kind of crowd, you’re going to really get them nervous,” Mr. Sanders declared to thousands at a rally in Austin on Sunday.

Mr. Sanders’s rivals have rejected the premise that he will expand the Democratic Party’s base, saying he is too rigid in his worldview. “Senator Sanders believes in an inflexible, ideological revolution that leaves out most Democrats, not to mention most Americans,” Pete Buttigieg, the former mayor of South Bend, Ind., said in his concession speech in Nevada on Saturday.

As Mr. Sanders and his opponents prepare for the South Carolina primary on Saturday, The Times’s analysis of the first three states show some challenging signs for his goal of producing a surge in turnout. In New Hampshire, for instance, turnout increased far less in townships he won than it did in townships won by Mr. Buttigieg and by Senator Amy Klobuchar of Minnesota.