Sturm played as a youth in his native Germany, appearing with ESV Kaufbeuren in the German Development League (DNL) from 2011 to 2014.

On 30 June 2015, Sturm was announced to have signed with the Storm, while also committing to play collegiate hockey at Clarkson University of the ECAC.

His solid two-way play was noticed as he led the NCAA with a 61.7% face-off win percentage and was named the 2017–18 ECAC Best Defensive Forward.

Returning as co-captain for his junior season with Clarkson in 2018–19, Sturm led the team in scoring and assists with 45 points in 39 games.

On the back of his success with the Avalanche, Sturm as a free agent was signed to a three-year, $6 million contract with the San Jose Sharks on 13 July 2022.