This documentary, "Building Pharaoh's Chariot"
has convinced me that the injuries on the mummy of Tutankhamun are consistent with falling out of a moving chariot while shooting arrows. The making of an Egyptian chariot, re-discovering a lost art, is fascinating, too--but one must watch the show to the end to see what may have happened to the young king. The injuries to his body are on the left side. The archer did not shoot the arrows straight ahead over the heads of the horses but from the side of the car. He chose his target or targets by having the driver charge past them. In a battle charge, the driver would maneuver the team in a circular pattern so that the archer would only expose himself to the enemy for a short time until the driver circled back. Watching how it's done in the video is far more edifying than any explanation from me.
Meanwhile, the bowman uses the open left side of the front panel and its railing in order to gain purchase with one of his legs. However, if the right-handed shooter is somehow ejected from the car at a good rate of speed, he will almost certainly land on his exposed left side. Watch the film and see if you agree.