A week ago, Joe Biden’s campaign was struggling and potentially near collapse. Since then, he has completed one of the most dramatic political turnarounds in recent memory with a string of primary wins that showed him to be the only candidate so far to expand the Democratic electorate, a claim that his top rival, Sen. Bernie Sanders, had made for himself.

The reversal was the result of a combination of signaling from party insiders, beginning with South Carolina Rep. James Clyburn, and more organic decisions by millions of...