Lynn began his racing career in karting at the age of eleven with Andy Cox Racing. In 2008 after four years in Mini Max and JICA classes he switched to Ricky Flynn Motorsport for competing in KF2 category, finishing British KF2 championship on the sixth place in the series standings.

In 2009 Lynn made his début in single-seaters taking part in the Formula Renault UK Winter Cup for Fortec Motorsport, finishing tenth in the standings. He finished the main Formula Renault UK Series in the same position, winning the Graduate Cup by scoring his first podium at the final race of the season at Brands Hatch.

Lynn continued his collaboration with Fortec for 2010 Formula Renault UK Winter Cup and the 2011 main series, becoming champion in both series, with three and twelve wins respectively. He also appeared in the Eurocup Formula Renault 2.0 at Hungaroring and Silverstone. His best result was second place from pole at Silverstone.

In the interval between 2010 Winter Cup and 2011 Formula Renault UK series he contested in Toyota Racing Series with Giles Motorsport, finishing ninth with a win in the first round of the series at Teretonga Park.
Lynn returned to New Zealand in January 2013 to contest the series again; this time with M2 Competition. 4 pole positions, 3 race wins, 9 podiums and 3 fastest laps gave him 2nd place overall and the highest placed international driver.
On 20 October 2011 it was announced that Lynn would represent Fortec Motorsports in 2012 for the fourth consecutive year in British Formula Three Championship. In addition he participated in selected Formula 3 Euro Series rounds. Lynn finished the 2012 British Formula 3 season in 4th place, with a race win at Silverstone and 5 podium places as well as 2 podiums in the Formula 3 Euro Series.
In November 2012 Lynn travelled to Macau to compete in the prestigious 59th SJM Macau Grand Prix. Lynn earned pole position for the qualification race, the first rookie to do so since 2006. In the race itself Lynn finished on the podium in third position and the highest placed rookie driver overall.
On 15 November 2012 it was announced that Lynn would be moving to Prema Powerteam and racing in the FIA European F3 Championship for the 2013 season. Lynn achieved 3rd overall in the championship and highest placed rookie driver, with 3 race wins, 14 podiums and 5 pole positions; including triple pole position at his home round at Brands Hatch.
In November 2013 Lynn returned to Macau to compete once again in the Star River Windsor Arch 60th Macau Grand Prix with Theodore Racing by Prema, a collaboration between SJM Holdings, Teddy Yip Jr's Theodore Racing and Prema Powerteam. Theodore Racing had last appeared at the Grand Prix in 1992 and was celebrating 30 years since they had won the event with Ayrton Senna at the wheel. Lynn won the qualification race to give him pole position for the main race and went on to dominate the main race from start to finish; becoming the first Briton to win the Grand Prix since 2007.

Lynn will compete at the GP3 Series in 2014 for Carlin Motorsport as a member of the Red Bull Junior Team programme. He won on his debut at the Circuit de Catalunya supporting the 2014 Spanish Grand Prix, starting from pole and setting the fastest race lap to take the maximum points from the race. In the sprint race he finished 18th, but still led the championship after its first round. Lynn won again at the Red Bull Ring in Austria, setting pole position before securing a comfortable win over his team-mate, Emil Bernstorff. But again he was out of luck in the reverse grid race two, finishing 20th after picking up a lap-one puncture. The Briton was disappointed to miss out on a home win in the third round at Silverstone in the UK, but second in race one and a fighting sixth in race two helped him to maintain his championship lead. In round four at Hockenheim in Germany, Lynn consolidated that lead by finishing second and third in the two races. He then recorded two fourth-placed finishes at the Hungaroring to extend his title advantage to 31 points over Richie Stanaway heading into the summer break.

Having won the GP3 Series title in 2014, Lynn was rewarded with a day of testing with GP2 team ART Grand Prix. He followed this up with a day of testing with Carlin. In early January, Lynn, along with fellow Red Bull Junior Pierre Gasly, signed with DAMS in order to win the GP2 Series crown in 2015

Following his win of the GP3 title, it was announced Lynn would partake in the post season testing, driving for Lotus F1. On 28 January 2015, Lynn was confirmed as the new development driver for the Williams F1 team, thereby severing his ties with Red Bull.