Under party rules, most delegates are awarded proportionally to Mrs. Clinton and Mr. Sanders based on their shares of the vote in congressional districts. The most Democratic-leaning districts are accorded the most delegates; in many places these are majority black and Hispanic districts, and Mrs. Clinton is far more popular with those voters than is Mr. Sanders.

“Support from African-Americans is going to be the key for Secretary Clinton across the South, and South Carolina is a good indication of that,” said former Gov. Richard Riley, who was the secretary of education under President Bill Clinton and is a supporter of Mrs. Clinton.

Mrs. Clinton’s strong support among black voters will serve as the biggest roadblock to Mr. Sanders’s chances for a surge in the weeks ahead. Clinton advisers believe she will trounce Mr. Sanders in Alabama, Arkansas, Georgia, Tennessee, Texas and Virginia, which have contests on Tuesday, and, in doing so, move even further ahead of him in their race to capture the 2,383 delegates needed for the Democratic nomination. About 880 Democratic delegates are at stake on Tuesday, the largest number on any single day during the primary season.

Mrs. Clinton also has backing from 453 superdelegates — party officials whose support counts toward the nomination — while Mr. Sanders has support from 20. Superdelegates can switch candidates at any time, and their support does not become official until the Democratic convention in July.

Mrs. Clinton edged ahead of Mr. Sanders among white voters, although he prevailed with white men, according to exit polls by Edison Research. Mrs. Clinton lost among white voters in the last two states, New Hampshire and Nevada, but South Carolina allies like former Gov. Jim Hodges had said they were confident that her message about creating economic opportunities would appeal to white voters even though it was often targeted to black voters.

But in interviews last week, several white Democrats were sharply critical of Mrs. Clinton and said they did not trust her, chiefly because of her use of a private email account and the American deaths in Benghazi, Libya, when she was secretary of state. While exit polls showed Mrs. Clinton and Mr. Sanders running evenly among voters who valued honesty and trustworthiness more than any other candidate qualities, some Clinton allies are deeply concerned that she could struggle with white men in the general election, and Republicans like Mr. Trump have made clear that they intend to sweep white Americans.