Back lights flash front lights don't when flasher button pulled. Bad flasher? Do the turn signal and 4 way flashers use the same flasher unit...(for troubleshooting)?
Are you saying this happens when you use the hazard switch, turn signal switch or both? There should be a separate hazard and turn signal flasher.
Technically the hazard flasher overrides the signal flasher by tying the left and right together. The front and rear are controlled at the contact in the turn signal switch.
Start by swapping the hazard flasher with the turn flasher and compare results, but they are cheap so I would just replace them both. Many of the signal issues I repair are caused by the hazard switch. It is rarely used and the contacts oxidize thus losing contact.