Two attackers, a police officer and a court worker are being reported dead.
Officials blamed Kurdish militants for the attack. A third attacker is reportedly still being sought.
The attackers drove a car to the courthouse entrance, sparking an exchange with police and then detonating the car bomb.
Other of people were injured in the explosion, some critically.
Some of Turkey’s big cities have been targeted recently both by ISIS and by Kurdish militants.
Turkey launched a military operation in Syria last year to push back ISIS and Kurdish forces from the Turkish border.
ISIS had said it was behind last weekend’s Istanbul club attack that left 39 people dead.
Izmir’s Governor Erol Ayyildiz the attackers were armed with Kalashnikov rifles and grenades.
Deputy PM Veysi Kaynak said that, judging by the weapons found following the raid, a much larger attack was being planned.
Images from the scene showed two cars ablaze. Erol Ayyildiz said that the second had been destroyed in a controlled explosion.
Reports of the number of people injured in the car bomb blast ranged from five to 11.
President Recep Tayyip Erdogan, speaking at the opening of a metro line in the capital, Ankara, said that Turkey was “under mutual attack by terrorist groups and they want Turkey to be brought to its knees”.
The Turkish president said: “They won’t be able to set people against each other. They couldn’t destroy our unity.”
No-one has yet said they carried out the Izmir attack but the governor said initial findings pointed to the involvement of Kurdish fighters from the outlawed Kurdistan Workers’ Party (PKK).
The PKK is fighting for an independent Kurdish state within Turkey.
It has carried out many attacks on Turkish security forces, particularly in the south-east.
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