The Independent Office for Police Conduct (IOPC) is appealing for witnesses to the collision, which happened in Lilycroft Road, Manningham, on Monday afternoon.
A West Yorkshire Police car hit a woman crossing the road at around 4.45pm.
It happened near the Cliffe Road junction and the marked police car was travelling towards Girlington.
The woman, who is believed to be in her 60s, was airlifted to hospital .
Disturbing footage of the incident has been posted online.
An IOPC spokesperson said: "Anyone who may have witnessed the incident is asked to contact our investigators via [email protected] or 0800 096 9076."