Sounds like the voltage to the turn signals is dropping when there is a heavy electrical load. That could be the battery, alternator, or just old, resistive connections. I think I'd get onto it with a voltmeter and see where the voltage is getting lost. Start at the battery, and if its voltage drops significantly when you turn on the lights, the battery is weak or its not getting charged well. The latter might be the charging system or maybe you just aren't driving enough to keep the battery charged.
If that checks out, it's time to start tracing through the circuit and seeing where the voltage is dropping, and go from there. Might be several places.