Media playback is not supported on this device
Guy, 20, took third behind winner Dutchman Sebastiaan Verschuren in the 200m freestyle after missing out on the men's 400m final.
Tutton, 19, claimed bronze in the 100m women's breaststroke race won by world record holder Ruta Meilutyte.
"That race was amazing," Tutton told BBC Sport. "I'm just over the moon."
Meilutyte triumphed in one minute 6.17 seconds, ahead of Iceland's Hilda Luthersdottir.
The Plymouth-based Lithuanian was winning back at the pool where she took the 2012 Olympic title aged 15.
Tutton edged out Irish swimmer Fiona Doyle for third in 1:07.50, with Tutton's team-mate Molly Renshaw in seventh.
Media playback is not supported on this device
In the 200m freestyle, Verschuren won gold in a time of 1:46.02, with Serbia's Velimir Stjepanovic second in 1:46.26 and Guy finishing in 1:46.42.
"For me, that was a great swim and a season's best," said Guy. "Rio is going to be a real scrap to the finish line."
Britain's Max Litchfield and Dan Wallace were seventh and eighth in the men's 200m individual medley final won by Greece's Andreas Vazaios.
Scot Ross Murdoch breezed through to Thursday's 200m breaststroke final with a winning time of 2:09.72.
England's James Guy and Welsh swimmer Chloe Tutton won bronze medals at the European Championships in London.