The Health Ministry said in a statement that the accident took place on a main road linking Beni Suef with the capital, Cairo.
The driver of the pick-up truck, which was carrying poultry, told local media that he fell with his vehicle from the bridge but managed to get out of the vehicle quickly, then the bus fell over his vehicle.
The driver is currently being treated in a hospital.
Hospital sources said that some of the injured are in a serious condition.
Egypt has one of the world's highest accident rates owing to careless driving and poor road and vehicle conditions.
The World Health Organisation (WHO) ranks Egypt as the world's tenth worst country in road accidents, after India, China and the US.
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