CEDAR RAPIDS, Iowa—In both subtle and not-so-subtle ways, the sprawling field of Democratic presidential hopefuls made clear Sunday that Joe Biden has a large political target on his back.

Following a week in which Mr. Biden stumbled as he reversed his stance on abortion funding, several 2020 candidates sought to contrast themselves with the Democratic front-runner at the Iowa Hall of Fame celebration, one of the party’s annual fundraisers. The heightened combativeness served as a preview to the first primary debates late...