A Sydney man allegedly raped a young woman he met through an escort agency before going on to kidnap and rape a 13-year-old girl from the city's west five days later.

The alleged sex offender's time on the run came to a dramatic end on Saturday when his car ploughed into a bus stop on a Bondi street during a car chase with police.

Detectives had been searching for Mustafa Kayirici, 26, since June 19 after a 22-year-old woman was allegedly sexually assaulted and robbed in the CBD.

Five days later, on Friday, he allegedly kidnapped a 13-year-old girl in Parramatta. It is understood Mr Kayirici did not know the teen but managed to lure her to him.