The crash, near the Nile Delta city of Damanhur, 160 kilometres (100 miles) north of Cairo, also injured 18 people, some of them seriously, police and hospital officials said.
Medics were not immediately able to say how many of the dead were children because the bodies were so badly burned after the vehicles burst into flames.
The fire completely gutted the bus which had been transporting the teenagers to school. Scorched text books were scattered near the wreckage, shown in footage aired by Egyptian television.
Medics said three charred bodies, including that of a police officer, were pulled out of a sedan which was also involved in the crash.
Provincial governor Mustafa Hadhud told Egyptian television that the bus had skidded after torrential rains struck the region.
One of the pupils who survived the crash said that the bus had arrived late and that the driver had explained "there had been a problem" with the vehicle.
"I was sitting in the back of the bus when the accident happened, and I jumped out of a window," the child told the private Egyptian CBC Extra in a telephone call from hospital.
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Roads in Egypt are often poorly maintained and traffic regulations little enforced.
On Sunday, 11 female university students were killed in a collision in the south of Egypt.
That accident sparked protests by fellow students in Sohag province.
Road accidents are responsible for an average of nearly 12,000 deaths a year in Egypt, according to the World Health Organization.
Some of the deadliest accidents have sparked protests and accusations of government negligence.
After Wednesday's crash, President Abdel Fattah al-Sisi ordered the prime minister to visit the scene.
He also gave instructions for the injured to be treated in military hospitals.
A cabinet statement promised "decisive measures to confront road accidents."