Harvey’s parents operate a steel-fabrication business in Lincoln, and he noted that without racing, he might have become involved in the family trade. In 2016, he briefly worked with his father as a laser cutter operator before deciding to move to America to continue pursuing a career in racing.
Harvey began his formula racing career in the 2009 Formula BMW Europe season with Fortec Motorsport, finishing seventh overall with a win at the Masters of Formula 3-supporting round at Zandvoort and two poles at Zandvoort and Monza. In 2010, he finished the year as vice–champion after an intense battle with Robin Frijns. Harvey entered the final round of the championship with a seven-point lead over Frijns, and extended his lead during the final round by taking the pole position for both races; unfortunately, he was pushed out of the track in the first race by DAMS driver Javier Tarancón.
In 2011, Harvey moved into the British Formula 3 Championship with Carlin, achieving one victory and four podiums in his first season. For 2012, he set his sights on winning the championship, opting to extend his commitment with Carlin. After seven wins amassing a total of twelve podium finishes, Harvey became the 2012 British Formula 3 champion. He then signed with ART Grand Prix to race in the GP3 Series in 2013, finishing fifth overall with two race wins.
For the 2014 season, Harvey moved to the United States and signed to race for Schmidt Peterson Motorsports in the Indy Lights series. He finished the season in second place in the championship standings, after collecting four wins and ten podiums in fourteen races. As a consequence, he was awarded the British Racing Drivers' Club's Earl Howe Trophy in recognition of being the top-performing British driver in North American motorsport. Harvey returned to Schmidt Peterson for the 2015 Indy Lights season, collecting two wins, six second-place finishes, and twelve top-fives in sixteen races, which put him runner-up in the overall standings, behind Spencer Pigot.
Harvey made his IndyCar debut at the 2017 Indianapolis 500, driving for Michael Shank Racing in association with Andretti Autosport, but finished 31st after contact with debris from Conor Daly's car on lap 65. In 2021, Harvey and Meyer Shank Racing ran a full schedule, with Harvey frequently a Q2 or better qualifier throughout much of the race weekends. On 11 October 2021, it was announced that Harvey would race for Rahal Letterman Lanigan Racing in 2022.
In 2025, Harvey was announced as a pit reporter for the 2025 IndyCar on FOX broadcasts, alongside Kevin Lee and Georgia Henneberry. He also became a part owner of his hometown football club, Lincoln City, in September 2025. On January 17, 2025, it was announced that Harvey would be attempting to run the Indianapolis 500, driving the No. 24 INVST Chevrolet for Dreyer & Reinbold Racing and Cusick Motorsports.
Harvey is close friends with fellow IndyCar driver, Conor Daly.
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