Thirty-four bus passengers, mainly schoolchildren and college students, were also injured.
The accident occurred near the unmanned railway crossing near Gopal Nagar railway station. The overcrowded private bus (en route 20) was on its way from Chakdha station to Bongaon.
According to locals, Amit, who was learning to drive a bike, was being instructed by his pillion riding father when the bus, approaching from the opposite direction, knocked them down along with their vehicle.
The bus fell into a roadside ditch after hitting the bike.
While Amit was crushed under the wheels of the bus, his father, Ashok Das, suffered severe injuries on his limbs and head, eyewitnesses said.
No one has been arrested till evening, said the police.
The injured passengers were taken to the local Bongaon hospital and some of them had to be shifted to Kolkata after their condition worsened.
Most of the injured are students of Giribala Ballika Vidyalaya of Bongaon and Bongaon College.
In a separate incident, two persons were killed in Barasat, North 24-Parganas today.
A private bus, which was heading towards Jagulia from Barasat town, fell into a roadside ditch killing two people after its driver lost control.
About 40 passengers were injured in the accident, 15 out of them suffered severe injuries and were admitted to the Barasat Hospital.
"Bodies of the victims, a male and a female, have not been identified yet," said Barasat police.
The driver and his helper were absconding, added the police.