Bridges joined the Robins from Sunderland in the summer, but has netted just once in 13 appearances for the club.
Carlisle boss Paul Simpson said: "He's a player that I tried to sign in the summer when I heard he was available, but Bristol City got him.
"We are delighted to be able to bring him here now as he is a very experienced player who can change any game.
"We're looking forward to working with him and I know he's looking forward to getting down to business with us."