[Lemans64]

Do the tail lamps work or brake lamps work after replacing the fuse, is there any funny function in the lights on rear of car or front for that matter.
Bad ground would do weird things as it is looking to ground out thru other devices causing low voltage on other devices. But should not blow a fuse.
also Blowing both turn signal fuse and brake light fuse would lead me to thing your have an issue with rear brake light, turn signal wiring or Bulb.
Yes could be just a bad Bulb. Replace the turn signal fuse and check to see when the fuse blows, right or left turn? Or just driving and not using the signals.