It happened just before 6:15 a.m. on the Exit 1A ramp from the northbound Parkway to the westbound Bruckner Boulevard, in the Throggs Neck section.
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It appears the bus driver failed to navigate the turn, lost control and overturned on the ramp.
Video from NewsCopter 7 showed the bus on its side and a number of adults lying on the grass near the bus.
Crash involving school bus in Bronx leaves several injured
A total of 37 adults - 36 passengers plus the driver - were on board at the time of the crash.
Three of the passengers were hospitalized in serious condition, while 33 others were checked out at the scene before being taken to area hospitals to be evaluated.
The driver refused medical attention.
New York City Mayor Eric Adams says the bus was carrying drivers for the city's Summer Rising programs.
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"Families should expect that some bus routes this morning and afternoon will be delayed, and schools will be reaching out to families whose routes are impacted," the mayor said in a statement. "Families experiencing delays should contact the transportation coordinator at their Summer Rising site or contact the DOE's Office of Pupil Transportation call center at 718-392-8855."
The bus company said it was able to cover half of the 34 routes affected by the accident and planned to double up on the remaining routes Thursday morning.
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The company said it expected to be able to cover all the afternoon routes.
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