State media reported that a policeman and a civilian were also killed, the BBC said.
Reuters reported that at least 10 people were wounded in the blast, according to state media.
Sky News quoted Izmir’s governor as saying that the car stopped outside the city’s courthouse.
Erol Ayyildiz said: “A clash erupted after our police officers wanted to stop a vehicle at a police checkpoint in front of Bayrakli courthouse.
“In this clash, terrorists detonated the car bomb while trying to escape.”
He said that evidence so far pointed responsibility towards the PKK (the Kurdistan Workers’ Party).
Sky News said that a rocket launcher, automatic rifle and eight hand grenades were seized.
The BBC reported that Turkey’s deputy prime minister Veysi Kaynak said that it appeared that a much larger attack was being planned, judging by the weapons found following the raid.
This latest incident comes just days after a gunman opened fire in a nightclub in the Turkish city of Istanbul on New Year’s Day, killing 39 people.
The so-called Islamic State terror group (also known as Isis, Isil or Daesh) claimed responsiblity for that attack, although the gunman remains on the loose.