The incident occurred at around 8pm tonight.
A Metropolitan Police spokesman said: "A car is believed to have been shot at by someone on a motorcycle.
"A man in his 20s has been taken to hospital. His injuries are not thought to be life-threatening.
"Three people have been arrested in connection with the incident."
The suspects are being questioned at various police stations.
The car came to a stop on the bridge, police said.
The bridge's southbound carriageway has been closed while detectives examine the scene.