Kurdish separatists, Islamist militants - including al Qaeda - as well as groups on the far left and right have all carried out bomb attacks in Turkey, but there was no immediate claim of responsibility for Friday's blast.
Turkish Interior Minister Idris Naim Şahin said security forces had followed the car from Göksun district in Kahramanmaraş province to Pınarbaşı - about 100 km (60 miles) - after it passed a checkpoint in the road without stopping.
Police opened fire as it passed the police headquarters in the town of Pınarbaşı and the bomb went off, Şahin said. Pınarbaşı lies east of the city of Kayseri, which is about 325 km (200 miles) southeast of the capital Ankara.
It was not immediately clear why police did not stop the car sooner, and Şahin said investigators did not think, as yet, that the target was the police station.
"From what can be seen inside the vehicle, there are two suicide bomber militants inside. They are of course in pieces and the car is destroyed. This is a sad event," Sahin told reporters from the Aegean city of Izmir.
"We were not expecting the terror organisation to stop. We have seen how they continue to carry out crazy acts," Sahin said, using a common term to describe the militant Kurdistan Workers Party (PKK), who are waging a war against the state.
Şahin said one policeman had been killed in the explosion and that 18 others had been wounded, eight of them seriously.
Turkish media earlier reported that the car had tried to drive into the police station moments before the blast and that security forces were hunting for a third militant who had managed to escape.
A civil servant at the Pınarbaşı district governor's office, said: "We heard a huge blast even though our office is 2 km away from the police station. We saw a big cloud of smoke rising but we don't know about the number of dead or injured."
"We heard that it was a car which attempted to drive into the police station and then an explosion occurred," Hasan Gümüş told Reuters by telephone minutes after the explosion.
Television footage showed frantic scenes around the police station with fire engines and ambulances rushing to the scene.
There was no immediate claim of responsibility. Bomb attacks in Kayseri are almost unheard of with most raids occurring further east in Turkey's predominantly Kurdish southeast, although militants have carried out isolated attacks in Ankara and western cities.
PKK terrorists vowed to step up their fight against the Turkish state last year and have launched a series of attacks against Turkish security forces, killing scores of people. Speaking hours after the blast, Prime Minister Recep Tayyip Erdoğan vowed to continue fighting the PKK.
"The fight against the separatist terror organisation will continue with determination, as it has until now. I wish health to those who are wounded, and God's mercy for our martyrs," he told reporters in Ankara.
More than 40,000 people have been killed in the PKK insurgency since the group took up arms against the state in 1984. The PKK is listed as a terrorist organisation by Turkey, the United States and the European Union.
Separately, 10 people, including one woman, were kidnapped by suspected PKK militants in southeastern Turkey this week, security sources said on Friday.
The sources did not know the whereabouts of the 10 villagers and said they had been unable to contact them since Thursday evening. Fırat, a news agency with close links to the outlawed PKK, said the abductions had taken place on Tuesday.
The PKK has not claimed responsibility for the kidnappings but the incident comes only two weeks after an official from the ruling AK Party was abducted in the region. Turkish security forces say they have launched a large-scale search operation for the missing official.