The bus drove into a kitchen shop on Lavender Hill, near Clapham Junction train station. Police did not immediately comment on the cause of the accident, and appealed to the public for any video footage of what happened.
London's ambulance service said it had treated 10 people for minor injuries, including the driver, and that three people who had been taken to hospital for further checks had been discharged.
The fire brigade also rescued two passengers who had been trapped on the top floor of the bus.
"This was a challenging incident in the early stages but crews worked hard to stabilise the shop and assist casualties alongside London Ambulance Service colleagues," said Ben King, station manager at the London Fire Brigade.
The bus was wedged into the shop front, with both its windows and those of the building smashed. Roads surrounding the scene were closed as emergency services continued to work to remove the bus.