An investigation has begun into what happened when the double-decker was in collision with a car in Clatterbury Lane, Clavering.
The 446 service, which was heading to Saffron Walden in Essex on Thursday morning, also collided with a low tree branch during the crash, which caused the damage to the roof.
Pictures from the scene show the whole of the upper deck torn off.
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Dawn Philpott, who had two children on the bus, told the BBC it was “some sheer miracle that everybody was able to walk away without fatality”.
The bus had been carrying pupils to the Joyce Frankland Academy and Saffron Walden County High School.
The academy said it understood that no serious injuries had been caused but that some children had been taken to A&E and minor injuries by their parents to have cuts and grazes looked at.
It added: “While most students appear to be OK, we know that this could have been a traumatic experience. Therefore we have made arrangements in school to provide a space for anyone who may be in shock or upset by the events.”
An Essex Police spokesman said: “We remain at the scene of a collision involving a bus and car in Clatterbury Lane, Clavering.
“The collision was reported at about 8.15am on Thursday 21 September, and closed the road while we and emergency services worked at the scene.
“Passengers on the bus were able to exit the vehicle and no serious injuries were reported from the scene at the time.
“The road remains closed while we continue to make the scene safe and drivers are advised to check before they travel and plan accordingly if their journey is affected.
“Some passengers have since attended hospital independently of the emergency services and five have sustained injuries considered to be serious, but none considered life-threatening or life-changing.”