This is a bit of a strange problem but the left turn signal on my '98 Saturn SW2 is acting a bit funny. When braking the blinker will either simply not work or work very slowly. When not braking it will usually work like usual but other times it will blink at 2x speed. I cant figure out what could be causing this problem only with my left blinker. Where would the division be in the electrical system that can allow for this strange isolation of behavior.
I'd check the ground strap/contact to ground from the blinker to the chassis first. Chances are there's a problem with it an it "grounds" via another bulb when you also apply the brake.
Using a multimeter, check the resistance from the ground on the bulb holder against the chassis first, then check blinker ground against both contacts for the brake light. Chances are that one of them will show a high resistance when it shouldn't and one to the brake light will show low resistance only with the brake applied.