A police spokesman said three women and 15 men were killed in the accident.
The paper reports 37 people were taken to 12 different hospitals and there were efforts to rescue two more people stuck on the bus.
The Hospital Authority said 19 people were in a serious condition.
The cause of the accident is not known.
Early reports said the number of dead stood at 19 but that was later revised down.
Reuters reported the roof of the bus was cut off to rescue passengers.
According to the SCMP, one passenger said the bus was going "much faster" than usual.
The injured passenger told the paper: "It was much faster than I normally felt in a bus.
"And then it was like the tyre slipped, and the bus turned. It was really chaotic in the bus. People fell on one another and got tossed from side to side."
Most of the victims were declared dead at the scene, and laid in white body bags along the side of the road.
A spokesman from the bus company, The Kowloon Motor Bus company, said HK$80,000 (£7,300) would be paid to each victim's family.
Photographs of the incident showed the bus on its side with the roof torn off.
The bus was travelling from Sha Tin race course. The fire brigade responded at 6.15pm.
The city's worst road traffic accident was in 2003, when a double decker bus collided with a truck and plunged from a bridge, killing 21 people.