The crash happened in Chapeltown in Pudsey at about 1.45pm on Monday, February 25.
Police said the passengers had only minor injuries.
Chapeltown and Upper Moor were closed while the police and fire service recovered the car.
West Yorkshire Police Leeds West said this in a statement on Facebook: "Neighbourhood officers have responded to a report of a Road Traffic Collision in Pudsey along with Paramedics and West Yorkshire Fire and Rescue.
“A number of vehicles were damaged and the cause of the collision is under investigation.
“The occupants of the vehicle received minor injuries. Chapletown and Upper Moor were closed whilst vehicles were recovered, however the road is now open for rush hour."