STOP. The relay in the fuse box is for flashing the lights when the SKREEM indicates "Lock" or "Unlock".
The turn signal/four way flasher is a function of the BCM (Body Control Module). If you want to troubleshoot this, download the service manual and follow the diagrams in the wiring chapter, turn signal section.

The hazards flash all four lights. This means you have enough current flowing so that the flasher works normally because there are four bulbs, even if they are not factory-spec, there is probably enough current for the flasher circuit to act normally.

The turn signal function is two lights, if not enough current is flowing, you get a fast flash, caused by:
1) One light is out.
2) If the lights are not the same as the factory bulbs, their current may be lower, too low to make the flasher work at the right speed.

Another possibility is that the flasher circuit in the BCM may have an issue - replacing the BCM would be the way out of that one.
But I think you have a bulb out OR you have out-of-spec bulbs.

In any event, you cannot troubleshoot this unless you understand how things work, and that means you should download the service manual.