Westbrook began karting in 1986 and progressed through Formula Vauxhall and Formula Opel Lotus before a lengthy career break from 1997 to 2002 caused by an inability to secure sufficient budget. He returned to motorsport by financing his own entry in the Porsche Carrera Cup GB, winning the title in 2004 with seven wins, 17 podiums, and eight pole positions. He was runner-up in 2003 and 2005, with his 2005 victory and pole totals still standing as series records.
Westbrook stepped up to the Porsche Supercup in 2006 with RT Morellato, finishing every race on the podium and clinching the title with two rounds to spare. He retained the championship in 2007 with HISAQ Competition, adding victories in Spain and Hungary. Following these back-to-back titles, Porsche signed him as a factory driver, making him the marque's first British works driver since Derek Bell. In this capacity he competed across the FIA GT Championship, the Le Mans Series, and the American Le Mans Series, becoming the only driver to win at least one race in all three categories during his factory tenure.
In 2009 Westbrook won the FIA GT2 Championship overall with Prospeed Competition, taking four victories in the Porsche 911 GT3 RSR at Silverstone, Adria, the Hungaroring, and Zolder. He made his first 24 Hours of Le Mans start in 2010 in the GT2 class with BMS Scuderia Italia.
In 2011 Westbrook joined Corvette Racing as a third driver for the American Le Mans Series endurance events and the 24 Hours of Le Mans. He added a full-season Rolex Sports Car Series campaign with Spirit of Daytona Racing in a Corvette Daytona Prototype from 2012. In 2013 he won the 12 Hours of Sebring as part of the Corvette Racing number 4 entry alongside Oliver Gavin and Tommy Milner, Corvette's first class victory there since 2009. He continued with the programme through 2014 and 2015, taking third in the inaugural United SportsCar Championship in 2014.
Westbrook joined Chip Ganassi Racing for the full 2016 WeatherTech SportsCar Championship season as co-driver of the number 67 Ford GT alongside Ryan Briscoe. Scott Dixon served as the third driver for endurance events including Le Mans, where the trio finished third in the GTE-Pro class in 2016. Westbrook and Briscoe contested full GTLM seasons from 2016 through 2019, finishing second in the Drivers' Championship in both 2016 and 2018. In 2018 the number 67 crew of Briscoe, Westbrook, and Dixon won the GTLM class at the 24 Hours of Daytona. The Ford factory programme was wound down at the end of the 2019 season.
With Cadillac Racing's entry into the Le Mans Hypercar category, Westbrook joined the programme alongside Earl Bamber and Alex Lynn in the FIA World Endurance Championship from 2023, driving a Cadillac V-Series.R.
Westbrook's record includes consecutive Porsche Supercup titles (2006–2007), the 2009 FIA GT2 Championship, the 2013 12 Hours of Sebring victory, second place in the 2016 and 2018 IMSA GTLM Drivers' Championship, and the 2018 Daytona 24 Hours GTLM class win. His longevity across multiple factory programmes — Porsche, Corvette, Ford, and Cadillac — marks him as one of the most consistent GT factory drivers of his generation.