The incident occurred while an open-top bus carrying Queen Beatrix and other members of the royal family was passing the crowd in the city, some 90 km southeast of Amsterdam.
The city mayor said four people were killed and 13 injured. It is not yet clear if the action, which is assumed to be deliberate, was an attack targeting the royal family.
"The car appeared unexpectedly. It hit between 10 or 15 people. The scene is terrible, blood is everywhere. Some of the people have been taken to intensive care," a photographer who witnessed the incident told national radio.
Video footage of the incident showed the damaged vehicle continuing to be driven at high speed after crashing through the crowd. A policeman on a bicycle narrowly escaped being hit by the car, which then slammed into a stone monument.
Dutch TV showed Crown Prince Willem-Alexander and his wife, Princess Maxima, standing on the bus and watching with surprise and horror.
Television coverage showed police taking a man from the black Suzuki and leading him to an ambulance. He was subsequently arrested.
Festivities were canceled in Apeldoorn and other towns.