The Detroit Tigers selected Turnbull in the second round, with the 63rd overall selection of the 2014 MLB draft. After signing with the Tigers, he began his professional career with the Gulf Coast League Tigers. After brief stints with the Gulf Coast League and Connecticut Tigers, Spencer spent 2015 with the Whitecaps, where he was 11–3 with a 3.01 ERA and 1.35 WHIP in 22 starts. He missed most of the 2016 season due to a shoulder impingement. Turnbull proceeded to spend the next two seasons in the minors before receiving his highly-anticipated call to the big leagues with the Tigers.
In 2019, the righty established himself as a Tigers rotation fixture in earning the start in the Comerica Park home opener and struck out a career-high ten Kansas City Royals in a 5-4 victory on April 4. A few weeks later, Turnbull earned his first career win in front of friends and family at Fenway Park as the Tigers beat Boston 4-2. In 2020, Turnbull posted a big-league best 3.97 ERA in 11 games, finishing with a 1.1 wins above replacement (WAR) score, ranking third-best among Tigers players.