Thiriet is the son of Claude Thiriet, the founder of the French frozen food company Thiriet. He competed in karting between 2003 and 2004 before beginning circuit racing in the single-seater disciplines.
Thiriet joined TDS Racing for the 2009 Eurocup Mégane Trophy, finishing fifth overall in his debut season. He remained with TDS for 2010, improved to second in the standings — behind teammate Nick Catsburg — and claimed two race victories while contributing to TDS Racing's teams' title.
Thiriet followed TDS into sportscar competition in 2011, contesting the Le Mans Series in the LMP2 class in an Oreca 03 shared with Mathias Beche and Jody Firth. The trio won the rounds at Spa and Estoril and finished fourth overall in the LMP2 drivers' standings. Thiriet also entered the 2011 24 Hours of Le Mans as part of the Luxury Racing entry, sharing a Ferrari 430 GT2 with co-drivers, but the car did not finish.
For 2012, when LMP2 became the premier class of the renamed European Le Mans Series, Thiriet and Beche competed under the Thiriet by TDS Racing banner. They won at Paul Ricard and at Road Atlanta during the Petit Le Mans, also taking second at Donington Park to secure the ELMS LMP2 title. At the 2012 24 Hours of Le Mans, joined by Christophe Tinseau, the trio finished eighth overall and second in the LMP2 class.
Thiriet was runner-up in the ELMS in 2013, winning two races with Beche. At Le Mans that year he did not finish, sharing the car with Ludovic Badey and Maxime Martin. In 2014, after TDS switched chassis — moving through a Morgan LMP2 and then a Ligier JS P2 — the highlight was Le Mans, where Thiriet, Badey, and Tristan Gommendy finished sixth overall and second in the LMP2 class. He finished eighth in the ELMS standings that season.
In 2015 he again claimed ELMS runner-up position, winning two rounds in an Oreca 05. The 2016 season produced a fourth ELMS vice-championship, with Thiriet sharing with Beche and Ryo Hirakawa and winning three rounds. Both the 2015 and 2016 Le Mans 24 Hours ended in retirement.
Thiriet spent his first full FIA World Endurance Championship season in 2017 with G-Drive Racing, sharing the Oreca 07-Gibson with Roman Rusinov and Alex Lynn. The entry finished 13th in the LMP2 standings; at Le Mans a collision forced retirement early in the race.
For the 2018–19 FIA WEC super season, Thiriet joined Signatech Alpine Matmut, sharing the Alpine A470-Gibson with Nicolas Lapierre and André Negrão. In June 2018 the trio won the LMP2 class at the 24 Hours of Le Mans, finishing fifth overall. A year later they won again at Le Mans in the season-closing round of the super season, finishing sixth overall, a result that secured the FIA Endurance Trophy for LMP2 Drivers and the 2018–19 WEC LMP2 drivers' championship for the trio. Thiriet finished the super season first in the LMP2 standings with 181 points.
Across nine Le Mans starts from 2011 to 2019, Thiriet scored two LMP2 class victories (2018 and 2019) and two further class podiums (second in 2012 and second in 2014).