A second car, a 2015 Maserati, was also involved in the crash but had minor damage and the driver was not hurt, the South Florida Sun Sentinel reported.
Norton was taken to an area trauma center.
Team officials said his injuries are not life-threatening.
But the Sun Sentinel says the car accident would be career ending and that Norton is listed in critical condition at Jackson Memorial Hospital's Ryder Trauma Center.
Paramedics performed the amputation at the scene so they could get him out of the vehicle, the newspaper reported.