Two women, aged 18 and 16, and a 20-year-old male were pronounced dead at the scene. A fourth man, also 20, died later of head injuries in hospital.
The crash, on the A6201 between Upton and Hemsworth at 00:30 BST, involved a bronze-coloured Nissan.
Two men, aged 21 and 22, who were in the car and were held on suspicion of dangerous driving remain in custody.
West Yorkshire Police said it was believed the Nissan had been travelling towards Hemsworth when it struck the rear of the quad bike.
Det Ch Insp Richard Holmes 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 at the scene and the road itself will not be re-opened until tomorrow."
He said the victims were all from the local area.
"Their families are currently being supported by specially-trained officers and we are not in a position to release any further details of those who have lost their lives at this time," he said.
Farmer Ben Kidson, who lives near the site of the crash, said he had heard loud cars late on Saturday evening.
"From about 11, for a good hour, I could hear cars some racing up and down the road.
"You do hear it now and again at weekends but it seemed to be a lot louder last night.
"It just seemed different and it continued until about 12:30 when I heard the police helicopter."
Police have appealed for witnesses to come forward.
Mr Holmes said they were also keen to speak to anyone who saw an unregistered quad bike being ridden around the Hemsworth area on Saturday evening.
"Similarly I would appeal for any witnesses who saw a bronze-coloured Nissan 350Z driven in the Hemsworth area on the evening of 26 September and up to 00.30 on the 27 September to come forward," he said.
Four people have died after the quad bike they were on was in collision with a car near Wakefield.