Defective flasher relay.
check light bulbs in both lights
Replace the signal flasher relay.
You are left with bad switches,corroded contacts, and broken wires.
Did you change the bulb in the rear? Yes. The hazzard lights use the same bulbs in the rear as the turn signals. They work with the hazzards on but not with the turn signal.
Go to AutoZone and look up your fuses in a book for the make and model.
check the fuse's first. then check the lamps [bulbs].....
Check sockets and grounds
Turn signals are mounted on a Chevy Cavalier just like a regular head lamp is. The difference between the units is the bulbs are smaller. Open the hood and find the lamp brackets. Unplug the old brackets and connect the new bulbs.
It could be the turn signal bulbs, some vehicles use separate bulbs for the hazard and turn signals. It could also be a bad switch or wires for the turn signals.
If the brake lights are not coming on in your car and you have changed the bulbs and the fuses with no results, you could also check the wiring in your '98 Chevrolet Lumina. The wires may be damaged or even broken.
Assuming all the bulbs have been checked, a weak flasher relay could do that.
bulbs are to small Check the fuses.