Colleen Calgaret, 20, is in intensive care after being hit while crossing Wellington Street early on Saturday morning.
Her sister, Jasmine Calgaret, said the family was desperate to know what happened.
Police would not confirm whether the car had turned left off the city’s Horseshoe Bridge, which is normally banned, or if the officer behind the wheel was responding to an incident.
“We’re in shock and we just want answers,” Jasmine said.
“I think we should be told more about the situation and what is going on about the investigation exactly.”
Another sister, Mariah Calgaret, said they felt left in limbo.
“We can’t say that anybody at this point in time was at fault,” she said.
Detectives want to hear from a group of people who they believe saw the crash.
The family is also asking witnesses to contact them directly.
They said they want to see any video themselves.
When asked, Police Minister Paul Papalia said: “As a pedestrian you’re the one who’s responsible for your own safety.”
Colleen’s grandmother, Colleen Ugle, says she just wants her granddaughter home.
She says she’s lost without her.
“Please somebody come forward, you must have seen it in town, the accident,” she said.
Colleen Calgaret’s partner and baby boy are by her bedside at Royal Perth Hospital.
