Turvey began karting at eight years old, winning the Junior Yamaha National Championship in 2000 and the Junior Rotax title in 2002. He entered formula cars in 2003 in British Formula Renault before graduating to Formula BMW UK with Team SWR. His 2006 Formula BMW campaign included five race wins and second place overall, and his performance at the Formula BMW World Final in Valencia as the highest-placed British finisher earned him selection for the McLaren Autosport BRDC Young Driver of the Year Award. He combined his racing career with an engineering degree at Fitzwilliam College, Cambridge, and on graduating in 2008 became the first racing driver awarded a Cambridge Full Blue — the university's highest sporting honour.
He competed in the British Formula 3 Championship in 2008, finishing runner-up behind teammate Jaime Alguersuari. In 2009 he raced in the World Series by Renault with Carlin alongside Alguersuari, winning one race and finishing fourth overall as the top rookie.
Turvey entered the 2009–10 GP2 Asia Series before competing in the 2010 GP2 Series full-time with iSport International. His backing from the Racing Steps Foundation expired at the end of 2010, preventing him from securing a budget for a second full season. He returned for selected rounds in 2011 for Carlin, beginning with Monaco where he finished in the points on his comeback before receiving a post-race penalty for jumping the start. He finished 25th in the 2011 championship.
In November 2010 Turvey drove for McLaren at the Formula One young drivers test in Abu Dhabi, setting the second-fastest time, 1.1 seconds behind Daniel Ricciardo. He subsequently joined McLaren as a test driver for the 2012 Formula One season alongside Gary Paffett.
In 2013 Turvey joined Jota Sport in the European Le Mans Series, taking pole position and winning his first start in the LMP2 Zytek at Silverstone. His highlight in endurance racing came at the 2014 24 Hours of Le Mans, where he was called up by Jota as a replacement after Marc Gené filled in for injured Audi driver Loïc Duval and vacated Jota's entry. Turvey and his teammates took the LMP2 class win.
Turvey made his Formula E debut for NEXTEV TCR at the 2015 London ePrix and became a long-term fixture in the series, racing continuously through to the 2021–22 season. His best result came at Mexico City during the 2017–18 season with NIO, where he finished second to claim his only Formula E podium. He earned a pole position at Mexico City in the 2016–17 season after Daniel Abt received a tyre pressure penalty, though he retired from the race with battery issues. After leaving NIO 333 at the end of the 2021–22 season, Turvey joined DS Penske as reserve driver and sporting advisor. In January 2025 he moved from his long association with McLaren to join Williams as a test and development driver.
Turvey's career represents the particular path of a highly educated British driver supported by a development foundation: strong enough to reach GP2 and earn a McLaren test role, yet ultimately finding the most sustained employment in the early years of Formula E. His academic achievement — an engineering degree from Cambridge alongside a competitive racing career — was recognised with the university's Full Blue, a first for the sport.