COLUMBIA, S.C.—Sen. Bernie Sanders’ insurgent 2016 presidential bid ran into a snag here after strong showings in Iowa and New Hampshire: His opponent Hillary Clinton trounced him in this early-voting state, foreshadowing the Vermont independent’s weakness in areas where the Democratic base is largely made up of nonwhite voters.

Three years ago, Mrs. Clinton beat Mr. Sanders in South Carolina with 74% of the vote to Mr. Sanders’s 26%. Nonwhite voters outnumbered whites nearly 2:1 in the state’s primary.

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