A special primary runoff that will test the power of President Trump’s endorsement and determine the political fate of a newly elevated senator is playing out on Tuesday in South Carolina, where early morning turnout appeared to be unusually strong for a runoff.

The election could bring a quick end to the congressional career of Senator Darline Graham, who filled the seat of her brother, Lindsey Graham, after his death last month. Ms. Graham, a political novice backed by Mr. Trump, is now running for a full term in the Senate and is trying to fend off Representative Ralph Norman, a hard-line Republican who has drawn support from a number of Republican lawmakers in Washington. The runoff winner will be the general-election favorite in November.

On Tuesday, Ms. Graham, 62, voted in Lexington, S.C., and planned to spend the afternoon with her family, said Abby Zilch, a campaign spokeswoman. She joined Mr. Trump at a virtual rally Monday night, the latest effort by the president to draw South Carolinians to the polls. Mr. Norman, 73, voted Tuesday morning in Rock Hill, S.C.

Runoff elections in South Carolina typically draw anemic voter turnout. But the race between Ms. Graham and Mr. Norman appeared to be upending usual trends, perhaps because of the president’s repeated interventions and the unusual circumstances surrounding the race.

As of 12 p.m. on Tuesday, five hours after polls opened, about 125,000 voters had cast ballots statewide, said Sydni Bain, a spokeswoman for the South Carolina Election Commission. The state also had relatively robust turnout for a runoff during the early voting period, Aug. 19-21, with some 64,000 ballots cast, according to election officials.

The state was on track to exceed turnout from the primary two weeks ago, when no candidate won a majority of the vote necessary to avoid a runoff, election officials said. At noon on Primary Day two weeks ago, about 100,000 people had voted, Ms. Bain said. That means early turnout was up about 25 percent on Tuesday. Typically, runoffs in South Carolina have lower turnout than standard primaries.

“If it keeps trending like this, it will be high,” Laura Booth, the elections director in Anderson County, said of Tuesday’s turnout.

For some South Carolina voters, it is the fourth time they’ve headed to the polls since June. The state also had a regularly scheduled primary in early June and a runoff for that election in late June.

The race on Tuesday is seen as a serious test of Mr. Trump’s influence in deep-red South Carolina. Mr. Trump romped to victory in the state in 2024, winning by more than 18 percentage points. But Republican voters have not shown total loyalty to the president. Twice in the past six months, candidates he endorsed have fallen in the state.

“There is some question as to whether there’s 100 percent support,” said Robert Oldendick, a political science professor at the University of South Carolina.

Mr. Trump has made a particularly heavy investment in Ms. Graham’s race. He visited the state to rally her supporters and pressured state officials to get behind her in a series of phone calls. His super PAC, MAGA Inc., invested some $800,000 in the race, nearly a third of its overall spending this year, according to campaign finance records.

In the virtual rally Monday night, Mr. Trump said South Carolina was “so important” to him. He described Ms. Graham as a “friend of mine” and a “great sister.”

“She loves this state more than anybody, and she loves our country,” Mr. Trump said. “She’s tough. She’s strong. And she’s the perfect person to carry on the legacy of her brother. Very different people. She is very much about that affordability. He was very much about foreign affairs.”

Ms. Graham is also backed by Senator Tim Scott of South Carolina. She won about a third of the vote in the primary two weeks ago, leading Mr. Norman by roughly eight percentage points in a crowded field.

Now Ms. Graham is seeking to overcome a headline-grabbing stumble at a TV debate last week. Responding to a question about Taiwan, she said that she was “not that informed on national security.”

Mr. Norman has assailed her over the debate gaffe, saying that South Carolina deserved a senator with a base-line understanding of foreign policy.

“She’s a propped-up candidate,” Mr. Norman, a fifth-term lawmaker, told Fox News in a recent interview. “You can tell that during the debates. This is an important job. There is no training on the job.”

Ms. Graham had scant political experience before her month in the Senate, though she previously led a state agency for blind residents. At the virtual rally, Ms. Graham said that Mr. Trump had “never served in an elected office before, and I think he’s done a darn good job.”

She has said she does not believe foreign policy expertise is a high priority for South Carolina voters.

But some appeared unconvinced that Ms. Graham was the better choice on domestic matters, either.

“I don’t think Darline Graham is the way to go for our state,” Hannah Benton, 43, said Tuesday morning after voting for Mr. Norman in West Columbia, S.C. Ms. Benton, a parent, said that she was focused on the cost of raising children and that she viewed Mr. Norman as “the lesser of two evils.”

But at Ballentine Community Center, in Irmo, S.C., where a steady of stream of voters flowed into the polls on Tuesday, Tom Coale and his wife, Karen Coale, both voted for Ms. Graham.

Their decision boiled down to two factors, they said. Ms. Graham was endorsed by Mr. Trump, they noted, and they supported her brother’s political career.

“I love Lindsey,” said Mr. Coale, clad in a black Make America Great Again hat. He added that he expected Ms. Graham to take positions similar to her brother’s. “She spent a lot of time with him.”

Emily Davies contributed reporting.