It appears to have been caused by a faulty front tyre, officials say, adding that the inquiry is ongoing.
The charter bus was taking pupils from Long Island to a band camp when it ran off the road and tumbled into a ditch near the town of Wawayanda.
Officials say 44 people were on the bus. They have been taken to six regional hospitals for treatment.
The crash occurred around 13:10 local time (18:10GMT) on Interstate 84, about two hours north of New York City. Police have closed the highway to allow work tob be carried out at the crash scene.
The bus was one of six that was chartered to take some 300 pupils from the Farmingdale High School marching band to a music camp in Greeley, Pennsylvania.
Photos show the bus on its side with broken windows and a ladder raised up to help rescue people trapped inside.
"Our hearts are with all who are impacted by this horrific situation," New York Governor Kathy Hochul said in a statement.
"At my direction, personnel from the New York State Police and the Division of Homeland Security and Emergency Services are onsite to assist local emergency response teams," the governor said.
CBS reports that most of the students on the bus were around 14 or 15 years old.
"I feel bad for them. I hope they feel better. I thought the tyre just came off. I didn't know they crashed and did flips and stuff," one student told the network.
"I usually take a bus," another pupil said. "I don't think I want to take a bus no more."
Some pupils said they have seen images of their injured friends on social media.
"My friend came up to me, showed me a photo, her face was all bloodied and stuff. I started crying," student Wenel Jen-Babtist told ABC News.
"We saw somebody posting on Snapchat, the bus flipped over, we thought it was a joke at first," student Sean Jacobs said. "We didn't even know it was serious. It's horrible."
The cause of the accident is under investigation by New York State Police.
A family reunification centre has been set up at the Orange County Community College.
The camp was due to run from Thursday to Sunday.
The uninjured students are expected to return home later on Thursday.