The Bahrain-Merida rider beat Italian Giulio Ciccone, who claimed the overall leader's yellow jersey at the top of La Planche des Belles Filles after a 7km ascent at an average gradient of 8.7%.
Overall, Ciccone leads France's Julian Alaphilippe, who wore the yellow jersey in the morning, by six seconds after the Deceuninck-Quick Step rider had to settle for sixth place. "I thought I had got away with it at the time and I didn't have any trouble when I got back into the race, no skin off or anything like that", Bevin said in a statement released by his team.
"When Movistar went and (Alejandro) Valverde were riding it was solid, I was feeling pretty good - I was just unsure as on those steep climbs I was expecting Richie Porte, Nairo Quintana and obviously Egan to jump up there, so I was hoping for it to be hardest all day before that". I think that sort of climb favours more the punchy, pure climber - say, Egan (Bernel), for one.
"It feels unusual, to have the yellow jersey on my back feels so odd." said Ciccone.
Thomas and Bernal, who dropped nine seconds to the Welshman, were part of an impressive Ineos team effort that controlled the pace over the most of the course as the peloton crossed seven mountains. Three of the overall contenders lost significant time as 2014 champion Vincenzo Nibali, Jumbo Visma captain Steven Kruijswijk and, to a greater extent, Frenchman Romain Bardet suffered on the slopes.
"I just wasn't up to it today, it's a bitter realisation", said Bardet, who has twice finished on the podium but who was the day's biggest loser. He took fourth place, two seconds ahead of fifth-placed Thibaut Pinot of France.
Thomas already lost five seconds to Colombian Bernal when he found himself at the wrong end of a split in the third stage on Monday.
Friday's stage is a long 230km run to Chalon-sur-Saone where a bunch sprint finish is expected.