Photographs on social media showed children in uniform walking from the blue doubledecker bus, which had a shattered windscreen.
Two people were rescued from the bus by firefighters using specialist cutting equipment at the scene near Carrowdore.
The incident happened at about 3.45pm. The Police Service of Northern Ireland described it as a “serious road traffic collision”.
The Northern Ireland ambulance service (NIAS) indicated most of the passengers had suffered “minor injuries if any”, such as cuts and bruises, with eight people requiring more serious medical treatment.
In a later update, it said most were well enough to be discharged at the scene, which is near the towns of Donaghadee and Newtownards.
A spokesperson said: “Following a road traffic incident this afternoon in the Carrowdore area, the NIAS declared a major incident, based on the number of resources required to attend the incident. Initial reports indicated that approximately 70 people were onboard the bus. This figure has been revised to 43 and a driver.”
Police are investigating the circumstances behind the crash, and Northern Ireland’s public transport provider, Translink, has said it is assisting with the investigation.
The bus had been carrying pupils from Strangford College to Bangor.
Dylan Lee, 12, a year eight pupil at Strangford integrated college, suffered minor injuries in the crash.
Speaking to the PA news agency at the scene, where he had been joined by his mother, Stacey Lee, he said: “I was on the top deck on the right-hand side. I was just sitting with my friend: he is fine, he just hurt his arm.
“The bus hit a post and it started to stall and went down this hill. It just started shaking. I closed my eyes and then I opened them and I was on the floor.”
He added: “The first thing I did when I got up was to phone my mum.”
His mother said Dylan had a lump and a cut on his head, adding: “I could hear him screaming that he had crashed and I could hear all the kids in the background screaming. It was awful.”
Dylan added: “Then there was this guy came down with a hammer and the windows were all being smashed so we could get out. I was crawling under stuff like railings and school bags and stuff.”
Stormont health minister, Mike Nesbitt, praised the response of emergency service crews and hospital staff.
He said: “This has been a shocking incident and my thoughts this evening are with all of those affected and their families.
“I want to express my heartfelt thanks to the emergency crews attending the scene and to the hospital staff caring for those who have been injured.”
Will Polland, a local SDLP representative, said concerns had been raised multiple times about the road where the crash occurred.
By Monday night the South Eastern Health Trust said the major incident response had ended but teams continued to treat patients at Ulster hospital.