A product of the FC Basel academy, Leutwiler was loaned to several other clubs in Switzerland for the duration of his time there, before joining Middlesbrough in 2012, from which he was released two years later and joined Shrewsbury Town. He helped the team win promotion from League Two. In August 2017, he joined Blackburn Rovers for an undisclosed fee. After making six league appearances in three years for the club, he moved to Fleetwood Town in September 2020, before joining Huddersfield Town in January 2021. Leutwiler has been called up to represent Switzerland at youth international levels, up to under-21. He later transferred his national allegiance to Canada, receiving his first call-up to the national team in October 2016.
Leutwiler joined League One club Fleetwood Town on 11 September 2020. He departed the club on 8 January 2021, following the expiration of his contract and joined Huddersfield Town on a deal to the end of the season. In June 2021, Leutwiler joined Oldham Athletic, signing a two-year deal.