Federal prosecutors accused the Turkish government of running a covert 2016 lobbying campaign in Washington aimed at pressuring the U.S. to hand over President Recep Tayyip Erdogan’s American-based nemesis, cleric Fethullah Gulen.

The disclosure came in an indictment unsealed Monday that accused two business associates of President Trump’s first national security adviser, Michael Flynn, of acting as agents of a foreign government, conspiracy and making false statements, in what prosecutors said was an effort to cover up their...