Hillary Clinton swept major primaries in Florida, North Carolina and Ohio on Tuesday, rebounding from her upset loss to Bernie Sanders in Michigan a week earlier and securing a political and psychological victory for her campaign.

The results were a significant setback for Mr. Sanders, who was counting on his fiery arguments against free trade to help him prevail across the industrial Midwest. He spent heavily trying to win Ohio, as well as Tuesday’s contests in Illinois and Missouri, but he came away with his presidential bid looking increasingly hopeless, since Mrs. Clinton is far ahead in amassing delegates needed to win the nomination.

Early Wednesday, The Associated Press declared Mrs. Clinton the winner in Illinois, too. Votes were still being counted in Missouri.

For Mrs. Clinton, Tuesday’s double-digit victories netted her so many delegates that her lead over Mr. Sanders is now about three times what Barack Obama’s was over her in 2008. On a personal level, too, she and her advisers were reassured that regardless of her Michigan defeat, her political arguments about jobs and the economy had potency in states that will be major battlegrounds in November.