The ignition and turn signals have no interconnection, one can't interfere with the other.
Power from the fuse comes to the flasher on a brown wire. From the flasher it goes out on a brown/white wire to the turn switch. From the turn switch to the signal lights.
Now find your flasher, unplug it. Now see if the left light lights up when the key is turned on.
If so the flasher is probably bad. Also use your meter on the 20 volt DC scale and test the voltage on the brown wire in the flasher socket. Is it the same as battery voltage?
Ok now make up a jumper wire so it can plug into the brown and brown/white wires of the flasher. Now try the turns. If everything is ok then they should come on but not flash. If so replace the flasher. Most any two prong flasher from the parts store should work fine. Just plug it in so the two prongs go into the brown and brown/white wires.
another thing to check is the turn switch itself. Remove the left switch housing from the bars. Look at the wires where they attach to the switch. Often they come apart, poor soldering at the factory.
Check the wires for the turns by the turns, they might have been pinched or cut.
Leo