Bridgman began karting in 2001 and obtained his racing licence the following year, entering the Zip Formula Series in 2002. He finished seventh in his debut season and then won the Zip Formula Great Britain championship in 2003 with five victories in twelve races. He also contested the Formula Renault UK Winter Series that season, finishing tenth.
In 2004, Bridgman stepped up to Formula BMW UK and won the inaugural championship, securing three victories and 230 points across twenty races. The achievement earned him a nomination for the McLaren Autosport BRDC Young Driver of the Year award and a BRDC Rising Star accolade. He tested with Carlin before opting to join Hitech GP for the 2005 British Formula 3 International Series.
His 2005 British F3 season was disrupted when he was forced to miss the Mondello Park round after sustaining burns to twenty per cent of his abdomen in a house fire. He finished the year fifteenth in the standings with 16 points, taking ninth-place finishes at Donington Park and the Nürburgring among his better results.
Bridgman moved to the United States in 2006 to compete in the Champ Car Formula Atlantic series with Jensen Motorsport. Over twelve rounds he scored a best result of fourth at San Jose, finishing sixteenth in the championship with 72 points. Returning to Britain for 2007, he entered Formula Palmer Audi with MSV, winning six races from twenty starts and four pole positions to claim the title. He also contested the Formula Palmer Audi Autumn Trophy, adding a further race win.
Bridgman joined the Porsche Carrera Cup GB in 2008 with Team Parker Racing, recording seven wins, eight pole positions, and thirteen podiums across twenty races to finish fourth in the championship in his debut season. In 2009 he took the title, securing eight wins, fifteen pole positions, and fifteen podiums. The championship was settled on the final day in a three-way battle with James Sutton and Tim Harvey, with Bridgman prevailing. He was awarded the BRDC Superstar recognition in 2010.
Bridgman stepped up to the international Porsche Supercup in 2010 with Schnabl Engineering and Team Parker Racing. Competing across ten rounds at Formula One venues, he finished fifteenth in the championship with 45 points, his best result a seventh at Barcelona.
Returning to British competition in 2011, Bridgman joined Trackspeed to race a Porsche 911 GT3 in the British GT Championship alongside Gregor Fisken. The pairing won one race at Brands Hatch and finished seventh overall in the championship. Trackspeed won the team title for the year.
After a hiatus, Bridgman returned for two rounds of the 2017 Blancpain GT Series Asia with HubAuto Racing, competing at Shanghai and Zhejiang. In 2018 he was drafted in to deputise for Jason Rishover at the opening round of the British LMP3 Championship with 360 Racing. His most recent competitive appearance was the 2020 British GT Championship season finale at Silverstone, where he raced a GTC class Porsche for Team Parker Racing alongside Karl Leonard.
Across his career Bridgman accumulated four championship titles: Zip Formula Great Britain (2003), Formula BMW UK (2004), Formula Palmer Audi (2007), and Porsche Carrera Cup GB (2009). His Porsche Carrera Cup GB campaign across 2008 and 2009 produced fifteen wins and twenty-eight pole positions combined — a record of dominance that remains notable in the series' history.