The accident occurred in Kampong Thom province in the central parts of the country just before dawn on Thursday.
The bus, which was en route from Siem Reap to the capital Phnom Penh, ended up in a river when it went off the bridge.
Another 20 people are reported to have been injured, according to police.
Around 40 people are feared to have been on the bus at the time of the accident, and divers are now searching for more possible fatalities.
A preliminary investigation into the incident indicates that the driver may have been sleepy when the bus drove off the bridge. It is unclear whether the driver is among the dead.
All those on board were Cambodian citizens.
