Police found the woman nearby with an injury to her leg. She did not require hospital treatment.
Footage posted on social media showed officers then pursued Mr Turnbull, who was holding the two dogs on a short lead along the canal, on Sunday afternoon.
The situation then appeared to become heated, as he was tasered to the floor and the animals were shot dead.
Mr Turnbull, 46, of no fixed address, was charged on Tuesday with being the owner of a dog that was dangerously out of control and being in possession or custody of a dog while disqualified from owning or keeping a dog.
He appeared in custody at Thames Magistrates’ Court.
Metropolitan Police said in a statement: “The charges follow a call to police just after 5pm on Sunday May 7 to reports of a woman being attacked by a dog in Commercial Road.
“Officers located a woman who reported two dogs had attacked her dog.
“Both dogs were destroyed by police at the scene.
“The owner of the dogs was arrested at the scene. Police taser was discharged. He was taken into custody and was later charged as above.”
The Met said its Directorate of Professional Standards conducted a “thorough review of the incident”, including available body-cam footage, and are satisfied that “there are no concerns around officer conduct”.
Commenting earlier on the incident, the force said officers “have a duty to act where necessary before any further injury is caused.”
“Officers attended the location where the aggressive behaviour of two dogs was of considerable concern and posed a significant threat to them.
“This is never an easy decision for any officer to take.”