Emergency services were called to the scene, close to the tube station, shortly after 2pm on Thursday (August 23).
Three people were taken to hospital and one person was treated as a priority at a major trauma centre, London Ambulance Service confirmed.
The incident is not being treated as suspicious and no arrests have been made.
According to the Metropolitan Police, the incident could have been caused by a medical episode.
"A car was in collision with pedestrians," a spokesman told getwestlondon .
"Officers are at the scene with colleagues from LAS and paramedics to assess the injuries.
"We were called at 2.05pm and the incident is not being treated as suspicious, no arrests were made.
"The driver stopped at the scene and none of the injuries are deemed to be life threatening."
A London Ambulance Service spokesman added: "We were called at 2.05pm today to Goldhawk Road, Shepherd's Bush, to reports of a road traffic collision.
"We sent three ambulance crews, a single responder in a car, an incident response officer. We also dispatched our hazardous area response team (HART).
"We treated four people at the scene. We took one of these to a major trauma centre as a priority, and the other three to hospital."
Jacob Dorrington was working at Sindercombe Social, located across the road from Brew Dog when the incident happened.
He told getwestlondon: "The car was going down Goldhawk Road and clipped one of those podiums you chain bikes to.
"He skidded around 20 meters down the road and crashed into people sitting outside Brew Dog.
"I saw paramedics wheeling a lady into an ambulance and a man was lying on the floor, he was being treated for around 20 minutes before being taken into an ambulance.
"The car is still outside, its boot is open and they’re rummaging around in it," he continued.
"It’s almost inevitable here, it’s so busy and there’s a lot of people driving who don’t really know where they’re going. It’s such a blind spot coming down the road to Brook Green."
Goldhawk Road reopened around two hours after the incident, however emergency services remained at the scene to assess the area.