WASHINGTON — In two courtrooms 200 miles apart on Tuesday, President Trump’s almost daily attempts to dismiss the criminal investigations that have engulfed his White House all but collapsed.

Mr. Trump has long mocked the investigations as “rigged witch hunts,” pursued by Democrats and abetted by a dishonest news media. But even the president’s staunchest defenders acknowledged privately that the legal setbacks he suffered within minutes of each other could open fissures among Republicans on Capitol Hill and expose Mr. Trump to the possibility of impeachment.

In Manhattan, Michael D. Cohen, the president’s former lawyer, admitted in court that Mr. Trump directed him to break campaign finance laws by paying off two women who said they had sexual relationships with Mr. Trump. And in Alexandria, Va., a jury found Mr. Trump’s former campaign chairman, Paul Manafort, guilty of eight counts of tax and bank fraud — the most significant victory yet for the special counsel, Robert S. Mueller III.

A president who has labored under the cloud of investigations from almost the moment he took office, Mr. Trump now faces an increasingly grim legal and political landscape. Mr. Mueller is methodically investigating whether Mr. Trump and members of his campaign conspired with a foreign power to win the election — and whether the president tried to obstruct the investigation from the White House. And the president is months away from congressional elections that could hobble the second half of his presidency.