West Yorkshire Police said Terrie Louise Kirby, 16, died along with her friends 18-year-old Alexandra Binns and Brandon Brown, 20, both from Upton.
Another victim is understood to be called Ryan Beal, aged 20.
Officers were called at about 12.34am on Sunday to a report of a collision between a bronze Nissan 350Z sports car and the quad bike on the A6201.
Alexandra Binns, Terrie Kirby and a 20-year-old man, who were all on the quad bike were confirmed to have died at the scene, while a second 20-year-old man later died in hospital of his injuries.
Two men aged 21 and 22, who were in the car, were arrested on suspicion of dangerous driving and have since been bailed pending further enquiries.
Det Chief Insp Richard Holmes from the homicide and major inquiry team, who is leading the inquiry, said: “Widespread investigations are continuing into what is a significant incident on the district’s roads which has now tragically resulted in the loss of four lives.
“Painstaking investigative work is continuing. The victims are from the local area and are two men aged 20 and two females aged 18 and 16.
“Their families are currently being supported by specially trained officers.”
He added: “We continue to appeal for witnesses to the collision and would also like to speak to anyone who may have seen a unregistered quad bike driven on the roads around the Hemsworth area in the evening of September 26 and up to 12.30am on September 27.
“Similarly I would appeal for any witnesses who saw a bronze-coloured Nissan 350Z driven in the Hemsworth area on the evening of September 26and up to 12.30am on the September 27 to come forwards.
“Anyone who has information is asked to contact the major collision and inquiry team on 101.”