Waqas Hussain is being sought by officers in connection with the collision on 17 July this year.
It happened on Bradford Road in Clayton, close to the junction with Hunters Park Avenue.
A West Yorkshire Police spokesman today said: "A 20-year-old woman suffered serious injuries when a Volkswagen Golf left the road and collided with a garage and tree.
"She has since been discharged from hospital."
Hussain, 22, is believed to live in Great Horton but also has links to the Clayton, West Bowling and Lidget Green areas of the city.
He is described as being of medium build, around 5ft 6in, and speaks with an Asian accent.
The spokesman added: "Officers have conducted extensive enquiries to locate Hussain, who is not thought to pose a risk to the public."
Anyone with information on his whereabouts is asked to contact the Safer Roads and Neighbourhood Support team on 101, quoting crime reference 13170327764.
Information can also be reported anonymously to the independent charity Crimestoppers on 0800 555111.