Why will the turn signals work only if flashers are on but the flashers will not work at all?

The relay that handles the turn signals is separate from the relay that handles the hazards.

Most likely you have a burned out turn signal relay. Most likely the relay is located in your fuse panel and is a very inexpensive item to replace.

turn signals and hazards have separate flasher cans both look the same and are generally in the fuse box replace the hazard flasher can. they cost about 2 dollars at auto store

HOUNDDAWG ADDS:

There may be a lamp with one of the turn signal elements burned out. If one is burned out turn signals may not work even though the lamp appears to be working when the lights are on.

Otherwise replace emergency flasher or possibly turn signal switch in steering wheel.

gaffer0118 adds:

I had the same problem on my 2003 Saturn L200. When I cycled the Hazard Signal/switch for about 30 seconds, both left and right turn signals once again worked!!