The collision yesterday (Monday) happened at 10.30pm in Newstead Lane, Fitzwilliam.
A grey Citroen Xsara collided with the barrier of a railway bridge in the road.
The driver of the Citroen, a man in his 20s, suffered serious injuries and was taken to hospital, where he remains today (Tuesday).
Police said the car was travelling close to a black Vauxhall Astra, and a 22-year-old man has been arrested in connection with the crash.
The Citroen had been driving along B6428 Hemsworth Lane, in the direction of Havercroft, when the collision happened.
A West Yorkshire Police spokesman said: "Police would like to hear from anyone who saw the collision or witnessed either of the Citroen Xsara or Vauxhall Astra driving in the area prior to the collision."
Anyone with information is asked to contact the Safer Roads team on 101, quoting log 2014 of September 25.