Republicans Ben Carson and Carly Fiorina and Democratic Sen. Bernie Sanders have gained significant ground in their parties’ presidential primary races in recent weeks, the latest Wall Street Journal/NBC News poll finds.

Former Secretary of StateHillary Clinton and continue to lead the fields for their parties’ nominations. But Mr. Trump is now essentially tied with Mr. Carson, and significant movement has occurred among candidates just behind them. Mr. Carson is backed by 20% of GOP primary voters, compared with 21% for...