Matt Gaetz, who faced a torrent of scrutiny over allegations of sex trafficking and drug use, abruptly withdrew his bid to become attorney general on Thursday in the first major political setback for President-elect Donald J. Trump since his election this month.

Mr. Gaetz has consistently denied the allegations, but his prospective nomination ran into trouble in the Senate, where Republicans were deeply reluctant to confirm someone to run the same Justice Department that once investigated him on suspicion of sex trafficking an underage girl, even though no charges were brought.

The collapse of Mr. Gaetz’s prospective candidacy just 16 days after the election appeared to be the earliest such failed cabinet pick in modern history and underscored the haphazard way that Mr. Trump has gone about assembling his new administration. He picked Mr. Gaetz almost on a whim last week without extensive vetting, knowing that allegations were out there, but essentially daring Senate Republicans to accept him anyway.

Mr. Trump moved quickly to replace Mr. Gaetz with another conservative favorite. By the day’s end, he announced on social media that he planned to nominate Pam Bondi, the former attorney general of Florida and one of his defense lawyers in his first impeachment trial, to be U.S. attorney general.

What remained unclear on Thursday was whether Mr. Gaetz’s withdrawal would embolden Senate Republicans to challenge other contentious cabinet choices, such as Pete Hegseth, the former Fox News host tapped for defense secretary, or Robert F. Kennedy Jr., the conspiracy theorist and vaccine skeptic selected for secretary of health and human services.

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