The 19-year-old’s motorbike and the bus collided at the Currie Street and King William Street intersection at 9.45am.
Heavy-duty equipment was used by emergency responders to lift the bus and remove the teenager from underneath the vehicle.
The crash was heard up and down Currie Street, prompting witnesses and business owners to attempt to free the rider.
“It was quite horrible, you know, like quite scary as well,” tech store owner Pardeep Kumar said.
“He was in so much pain, he was right underneath the back rear tyre.”
The rider, from Henley Beach South, was taken to Royal Adelaide Hospital with non-life-threatening injuries.
The bus driver was not injured in the accident.
Traffic was temporarily halted on Currie Street.