The 25-year-old was rushed to Leeds General Infirmary following the incident at Birstall Shopping Park at around 6.15pm on Saturday.
His condition was today described as critical but stable.
Police said that there was initially a collision involving two vehicles in the car park outside Marks and Spencers.
This resulted in the occupant of one of the cars, the victim, chasing a Seat Leon and then clinging onto the front passenger window as it drove off at speed.
The man, who is local to the Birstall area, was then in collision with a parked Nissan Navara and fell to the ground.
Three 17-year-old boys and a 51-year-old man, all from Bradford, have been arrested on suspicion of wounding with intent and released on bail.
Detective Sergeant Damian Roebuck, of Kirklees District CID, said: “We have already spoken to a number of witnesses at the scene but would urge anyone else with information about this incident who we have not spoken to already to please come forward.
“This offence has taken place in a busy car park and it was necessary to close the car park for a short period of time while we dealt with this incident. We would like to thank the many shoppers and shop workers affected for the patience they showed during this time.”
Phone Kirklees CID via 101 or to pass information anonymously to independent charity Crimestoppers on 0800 555 111.