Do all the lights work properly with the hazard flashers on? If not, try removing the right front bulb and then switch the turn signals on. Do the rear turn signal lights now flash at the same rate in both left and right sides?
If it does, the easy thing to try is to replace the bulb. This may or may not fix it at this point. If removing the bulb does not change the rate of flashing down to normal speed, and the rear lamp continues to flash at a high rate of speed, look in the bulb socket and at the base of the bulb for any signs of corrosion or damage. If you do not see anything unusual, I would suspect bad wiring in the right turn signal circuit. The front and rear turn signal lights are powered from the same circuit and get there power from the same source.
Does anything change if you have the headlights on and then turn the right turn signal on? If things then change with both systems on, check for bad grounding for that lamps wiring. Bad grounding can cause the power to back feed to a different circuit as it tries to find a place to ground. This can cause all kinds of weird things to happen. Easy things to find are when a circuit is either shorted out to ground or is fully open, as in a broken wire. The really hard electrical problems to find are the partially grounded or partially open wiring or connections. I would also recommend that if you have any LED bulbs in this lighting system for the turn signals, remove them and go back the the proper bulb for the car. At the very least, remove them for troubleshooting this problem as flaky LED bulbs can add to the weirdness factor in trying to diagnosis your problem.