Bridges was signed by Chelsea in July 1956 having been spotted whilst playing for Norwich and Norfolk Boys. He turned professional in May 1958 made his debut for the club against West Ham United in February 1959, and scored in a 3-2 victory. He first established himself in the Chelsea side during the 1961–62 season, and though the club were relegated, he nevertheless proved himself a prolific goalscorer as he found the net 20 times that season.
A prolific and versatile forward who could play in the centre or on the wing, Bridges was an important part of manager Tommy Docherty's re-structured Chelsea attack, alongside Bobby Tambling and Terry Venables. He helped the club gain promotion back to the First Division at the first attempt, and then challenge for the major honours. His most successful season with the club was in 1964–65, in which he scored 27 goals in 42 matches, as Chelsea won the League Cup but faltered late-on in their championship and FA Cup challenges.
It was during that season that Bridges, along with several other players, was punished by Docherty for breaking a curfew, after which the manager's relationship with his players deteriorated. Following one more season with Chelsea, Bridges was sold to Second Division side Birmingham City in May 1966 for a club-record £55,000. He continued his impressive scoring record at St Andrews, netting 47 times in just over 100 games, and helped the side reach the League Cup semi-finals in 1967 and the FA Cup semi-finals a year later. Following the latter defeat, he moved to Queens Park Rangers and then had spells at Millwall, Brighton & Hove Albion and Highlands Park in South Africa. He retired having scored over 200 career goals.