A former England Youth international, Carl began his career in his hometown with Sheffield Wednesday originally playing in a striker's role. He made his league debut whilst on loan to Barnsley before accumulating 32 league appearances and 4 goals for the Owls. He transferred to Manchester City but did not enjoy his spell there making just 5 appearances. A return to Yorkshire came when he was signed by Sheffield United in September 1989. 147 league games for the Blades brought Carl 8 goals and a switch to an attacking right back position which gained the attention of Norwich City manager John Deehan as a replacement for Ian Culverhouse .

Deehan paid £450,000 for Carl in July 1994 and he made his City debut at Chelsea on the 20th August 1994 in a 2-0 defeat. Injury problems limited his first team appearances to just 29 games (1 as sub) in a year that saw the Canaries relegated from the Premiership. However he did score the winner in the local derby clash with Ipswich Town. Two seasons in Division One were beset with injuries and disciplinary problems. In the early part of 1997, Bradshaw and Andy Johnson were involved in a late night fracas at Colney which resulted in Carl biting the hand of a taxi driver. In September 1997, he was sentenced to a brief spell in jail for this assault. Upon his release, his contract was terminated by Norwich City leaving his Canary record at 74 games and 3 goals. Wigan manager John Deehan took this opportunity to renew his relationship with Bradshaw and signed him on a free transfer. He made 28 appearances for his new club in his first season and scored once.

Carl returned to Carrow Road a year later to play for Wigan in their Worthington's Cup clash with Norwich but scored the only goal of the game in his own net to give Norwich a 1-0 victory. He was the second ex-Canary (after Stockport's Colin Woodthorpe )to score an own goal to benefit the Canaries that season.

In July 2000, with his contract expired at Wigan, Carl was linked with a move to Blackpool where he could team up again with Mike Milligan. However Carl signed a new contract on 18th July 2000. Despite playing an instrumental role in Wigan's successful 2000/01 season where they reached the Division Two play-offs, Carl was surprisingly linked on 13th May with a move to Division Three side Halifax. One only imagined that the new position has a coaching element to it as Wigan were in the Division 2 play-offs. However maybe the media in Halifax knew something after all as Wigan's failure in the play-offs resulted in necessary cuts at the JJB Stadium and Carl was one of those victims as his contract was not renewed. Carl signed for Third Division Scunthorpe on 11th July 2001. He was released by the Iron at the end of the 2001/02 season.

In mid-July 2002, Unibond First Division side Alfreton announced that Carl was to join them for pre-season matches and he completed a permanent switch to them towards the end of the month. In his first season with Alfreton, they won the Unibond First Division title and the Derbyshire Senior Cup.

Carl was released by Alfreton at the end of the 2004/5 season. He is now working in his native Sheffield as a qualified bricklayer, working on the city's Decent Homes programme that sees many properties receive extensive internal and external refurbishment. Carl said "I was never interested in staying in football after I stopped playing. I am more of a family man and I can't do with all the backstabbing in the game. I know some of these areas well and I know that this kind of work is something that is needed and that it will be of real benefit to people."