The road was closed at Limberlost, the Upton bypass and Vines Farm Shop and police accident investigators have been at the scene all day trying to establish what happened.
The road has been reopened to traffic using temporary traffic lights and the bus is still at the scene.
One student said: "I have never been so scared in my life. I feel so lucky to be alive and would like to thank the kind men who helped us escape."
A spokesman for First Hampshire and Dorset said: "The exact cause of the incident is not yet known but it will be thoroughly investigated.
"Our thoughts are with those who have sustained injuries as a result of this incident; we wish them all a full and speedy recovery.
“Our team attended the scene and are now working with the relevant authorities to determine the cause. Safety is our number one priority. All of our drivers are fully trained and hold all relevant qualifications before transporting passengers, as well as undergoing specific learning on each route.
"As is normal industry procedure, today's driver was being monitored by an experienced colleague as part of that process."
Kingston Maurward College Principal Clare Davison said: “The safety and well-being of our students is our top priority and obviously we are extremely concerned about this incident.
“We have set up a team at the college and we are liaising directly with the emergency services. We are talking to First Buses and will work closely with the police to understand how this occurred. We will also be offering additional support to the students involved.
“We are calling all parents and guardians of students affected and have set up a direct line for them to contact us. Anyone concerned can call 07894 751302.”
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