The son of a farmer, Menu began his motorsport career in single-seater racing, progressing through British Formula Three, the British Formula 3000 Championship (in which he finished runner-up in 1990), and the International Formula 3000 Championship in 1991. He switched to touring cars in 1992, racing a BMW in the BTCC, showing immediate promise before being injured mid-season in a quad bike accident at Knockhill.
In 1993 Menu began a six-year association with Renault in the BTCC, initially with the Renault 19 under GB Motorsport and then from 1994 onwards with the Williams-run Renault Laguna. The Laguna proved a consistently competitive platform. Menu finished second in the championship in 1994 behind Gabriele Tarquini, and again second in 1995. In 1996 he was second once more, 92 points behind champion Frank Biela's dominant Audi.
The 1997 season brought Menu's first BTCC title. Racing alongside Jason Plato under the Williams-run programme, he took 12 victories across the season — a record that was not equalled until 26 years later — and secured the championship with several rounds remaining. In 1998, with Nescafe as headline sponsor and a less dominant car, he won three races and finished fourth overall.
Menu switched to Ford for 1999 but the Mondeo was uncompetitive; he managed one win and finished eleventh. In 2000, however, Menu and the Mondeo came good. Competing alongside Anthony Reid and Rickard Rydell in a three-way team battle, Menu claimed his second BTCC title with six victories, becoming champion for the final year of the Super Touring regulations. He would be the last driver to win the BTCC in a Ford until 2023.
Menu made a one-off return to the BTCC in 2007, driving a Vauxhall Vectra for VX Racing at the Thruxton finale to assist Fabrizio Giovanardi's title bid. He returned more permanently in 2014 with Team BMR in a Volkswagen CC under the NGTC regulations, scoring a pair of podiums and finishing tenth in the standings. He remained with Team BMR after 2015 as a driving coach.
From 2001 to 2003, Menu competed for Opel in the Deutsche Tourenwagen Masters but struggled to translate his touring car credentials into results in the high-tech German series. He described the DTM period as "not very good" in a 2004 interview.
Menu joined the Chevrolet RML team for the inaugural 2005 WTCC season. Early years were difficult, but in 2006 he scored Chevrolet's first-ever outright win in an FIA world championship event at a wet Brands Hatch. He continued with Chevrolet as they developed through the Lacetti and Cruze generations, winning races across multiple campaigns. His most successful WTCC season was 2012: he won races at Spain, Morocco, the USA, Japan, and China, finished the year at Macau with a final-round victory, and ultimately claimed second in the drivers' championship by twelve points behind teammate Rob Huff.
Chevrolet withdrew from the WTCC at the end of 2012, ending Menu's primary programme. He subsequently raced in the Porsche Supercup for a single season before his competitive career wound down.
Menu was voted the fourth greatest touring car driver of all time in a 2005 Motorsport Magazine poll — a reflection of his sustained excellence across the Super Touring era. His record of two BTCC titles during the series' most competitive period, combined with his WTCC longevity, places him among the most accomplished touring car drivers of his generation.