ANAMOSA, Iowa—In the days surrounding Bernie Sanders’s heart attack on the campaign trail, his support dipped, at the same time as liberal rival Elizabeth Warren was surging into a near tie with Joe Biden for the lead. The Vermont senator’s chances for the Democratic presidential nomination seemed dim.

Just over three months later, many polls show Mr. Sanders at the front of the pack in both Iowa and New Hampshire. He raised more money last quarter than any of his Democratic competitors. His campaign says more than 10,000...