Turris' path to the NHL began when he made a big impression as a 16-year-old skating for Burnaby of the British Columbia Hockey League. He finished with 72 points (36 goals, 36 assists) in 57 games and was awarded the Bruce Allison Memorial Trophy as rookie of the year in the Coastal Conference.

The following season, he was named conference MVP after finishing second in the BCHL with 121 points (66 goals, 55 assists). During Burnaby's run to the Royal Bank Cup, Canada's national Junior A championship, in 2006, Turris had 26 points (12 goals, 14 assists) in 14 playoff games, including a hat trick against Yorkton of the Saskatchewan Junior Hockey League.

Turris was No. 1 on NHL Central Scouting's final ranking of North American skaters for the 2007 NHL Draft, where he was chosen by the Phoenix Coyotes with the No. 3 pick behind Patrick Kane (Chicago Blackhawks) and James van Riemsdyk (Philadelphia Flyers). After two full seasons with the Coyotes, Turris was traded to the Ottawa Senators for defenseman David Rundblad and a second-round pick in the 2012 NHL Draft on Dec. 17, 2011.

Turris excelled as a top-six center for the Senators and scored 27 goals in the 2016-17 season to help them advance to the 2017 Eastern Conference Final, where they lost in seven games to the Pittsburgh Penguins. Eleven games into the 2017-18 season, Turris was acquired by the Nashville Predators in a three-team trade that sent center Matt Duchene to the Senators and prospects to the Colorado Avalanche on Nov. 5, 2017. Turris was subsequently signed to a six-year, $36 million contract with the Predators.