GREENVILLE, S.C.—After a raucous week of campaigning, Donald Trump remains atop polls in the Republican primary field here as voting begins Saturday in the state that brags that it picks presidents.

In South Carolina, Mr. Trump is the top pick of 28% of likely Republican primary voters, according to a new Wall Street Journal/NBC News/Marist poll. But that is down eight percentage points from a January survey, and his 16-point lead in the earlier poll has narrowed to five points over Texas Sen. Ted Cruz’s 23%. Florida Sen. Marco Rubio was third, at 15%, and former Florida Gov. Jeb Bush followed close behind in fourth place, at 13%.

The crowded contest for second place may mean the primary here won’t significantly winnow the field before the Feb. 23 GOP caucuses in Nevada, or even by March 1, when a swath of mostly Southern states will weigh in on the race.

“I think this thing goes to Florida,” said South Carolina Sen. Lindsey Graham, a Bush supporter who was referring to that state’s March 15 primary. “I don’t see South Carolina as being a knockout punch for anybody.”

While all the candidates promised to soldier on, a broad Trump victory here would cement his status as the Republican-to-beat for the nomination and likely trigger cries from party heavyweights to rally around one alternative.