Turvey started his motorsport career after being taken to a karting event by his father who worked in car industry and wanted to try something new. He instantly took to it and convinced his family to let him race every week. Soon, his hobby became something more serious when he started taking part in races at his local track.

A few local club championships later and Turvey was told he should think of racing professionally. He made his single-seater debut in the British Formula Renault Winter Series and a year later in Formula BMW UK. The following year, without the backing to race the full season, the British driver had to miss the start of the Formula BMW season.

In 2006, he beat five other finalists to win the prestigious McLaren AUTOSPORT BRDC Award, which led to him making his Formula 1 test debut for the British team in 2009. He remains part of McLaren’s testing roster to this day.

In 2008, Turvey was runner-up to Jaime Alguersuari in British F3, which led to a season of Formula Renault 3.5, during which he scooped victory at Monaco. Despite clinching a win in GP2 Asia, he never had the same success in the main series and in 2012, Turvey moved into GT racing with McLaren. Outside of Formula E, Turvey raced in the Japanese Super GT series in 2015 and 16, as well competing in the 24 Hours of Le Mans and the World Endurance Championship for CEFC TRSM Racing in the LMP1 class.

Moving to Formula E in 2015, the Brit finished inside the top ten during his debut race. From there, Turvey remained with the Chinese team for a second season in 2015/16. Struggling through with an unpredictable car, his opportunities to shine were limited, but still he managed to make it through to both Super Pole sessions in the London doubleheader. His strong qualifying form continued into the 2016/17 season and he scored his maiden pole in Mexico where he led the race until a mechanical issue caused his car to stop on track. Into his fourth season in the series (2017/18), Turvey looked to clinch his maiden win in the Championship in Mexico after starting on Pole but it wasn't to be. The British driver settled for second behind Audi Sport Abt Schaeffler's Daniel Abt.

For the 2019/20 season, Turvey lines up alongside teammate Qinghua Ma as both drivers challenge for the title. With four full seasons behind him, Turvey will be hunting for his maiden win in the series.