Mrs. Clinton’s campaign says she will win over young voters when they learn more about her proposals to make college affordable and to combat climate change. “The difference for young voters and for all voters is the ability to get something done,” Robby Mook, her campaign manager said in an interview with MSNBC this week. “And what that takes is someone in Washington who can break through the gridlock.”

That argument won over Iris Brenner, 21, a student at Iowa State University in Ames, who supports Mrs. Clinton. “For me, it’s been, ‘Do I caucus for someone who is a little less exciting but who can get stuff done?’”

Privately, Mrs. Clinton’s supporters say that while being a youth icon has its advantages, the support of middle-aged and older voters is enough for her to capture the nomination. In the Iowa caucuses, she beat Mr. Sanders by 23 points among voters ages 45 to 64 and by 43 points among voters 65 and older, according to exit polls.

“They don’t have to be for me, I’m going to be for them,” Mrs. Clinton told CNN on Wednesday when asked about the generation gap.

The Sanders campaign said excitement among young people had been organic, although it has clearly worked to improve the candidate’s appeal on sites like Twitter, Reddit and Snapchat, and in text messages to supporters. “Young people are idealistic and they look at this country and say we can be much more,” Mr. Sanders told reporters Wednesday.

The discomfort, and, in some cases hostility toward Mrs. Clinton among young voters is striking. Some of them, feeling the pinch of economic hardship or the burden of college loans, suggest she is too cozy with big banks and corporate America. But polls also show they do not find her trustworthy.