Police and emergency workers were called to Highams Park in northeast London after the collision, which took place during the morning rush hour on Tuesday.
London Ambulance Service said medical staff treated 19 people at the scene, including the five who were taken for hospital treatment.
No details were given on how serious their injuries were.
Photos from the scene showed the front of the bus was damaged and the windshield shattered.
Eric Garip, the owner of a cafe at the scene, said he rushed out after hearing a "big bang" on his road.
He said he tried to help calm and move out some of the children who were on the bus but they "were all scared and crying".
"There were a couple of kids on the floor," he said.
"We were trying to take the kids out and they were panicking... Some parents were in the bus as well crying. I tried to calm them but it was a big shock."
Police have closed roads in the area as emergency services worked at the scene.
A police statement said the cause of the crash was under investigation.
London Mayor Sadiq Khan said his thoughts were with those affected by the collision.
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