A silver Ford Focus is understood to have clipped a kerb and flipped over in Harborne Park Road, near the junction of Denis Drive, in Harborne, at around 8.50pm on Wednesday night.
A woman in her 30s suffered serious injuries and remains in a critical condition in hospital, while a teenage girl also sustained minor injuries.
West Midlands Police said it received reports that two men fled the vehicle, and officers are carrying out enquiries to identify and trace them.
The car has been recovered and will now be forensically examined.
Detective Sergeant Paul Hughes, from West Midlands Police's collision investigation unit, said: "A woman has been left seriously injured and we are in the process of establishing the full circumstances around what happened last night.
"We have spoken to witnesses who say two men were seen getting out of the vehicle after it overturned and then fled.
"I would urge them to do the right thing and come forward to tell us what went on."
Two ambulances, three paramedic officers and the MERIT trauma doctor and critical care paramedic attended the scene.
A spokesman for West Midlands Ambulance Service said: "Crews arrived at the scene to find a woman, believed to be the driver of the car, trapped in the vehicle.
“Staff worked with colleagues from the fire service to safely extricate the woman so that a full assessment of the patient could be carried out.
“She was treated for potentially serious injuries at the scene before being conveyed to Queen Elizabeth Hospital Birmingham, with the MERIT trauma team continuing treatment on route.”
Anyone with information is urged to call West Midlands Police on 101, or email FL_COLLISION_INVEST@west-midlands.pnn.police.uk, quoting incident number 4062 of June 9.