WASHINGTON—This week, Republican Rep. Trey Gowdy inherited one of the most difficult jobs in Washington: leading Congress’s most powerful investigative committee charged with monitoring the workings of the federal government while a president of his own party is running it.

A former prosecutor from South Carolina, Mr. Gowdy, 52 years old, was elected by his Republican colleagues this week as the new chairman of the House Oversight and Government Reform Committee, the only congressional panel dedicated full-time to investigating...