Check the grounds and wires on that side. Sounds like you have corrosion somewhere.
As I understand it, the turn signal relay is a thermal relay - it uses current+resistance to heat a wire which closes the turn signal circuit, lighting up the lamps. Closing the relay cuts power to the resistance wire, cooling it, thus opening the circuit and the lamps go off. It continuously repeats this cycle to provide intermittent flashing turn signals.
For any given voltage, the more you increase the resistance, the less current you get. Corrosion increases resistance, and lowers current. Lower current will take longer to heat up the wire and close the thermal relay. This would slow down the rate of flash.
A very simplified diagram of a turn signal setup:
You know the flasher relay works for one side. So your problem lies somewhere after the relay. Maybe corrosion in the turn signal switch or wiring or grounds on the problem side.