If you want to have the lights on all the time but flash when you activate the turn signals, I would suggest that you use a simple relay circuit (either solid state or mechanical). Since you have both left and right turn signals, you would need two separate relays. You need to make sure that the relays have a "Normally Closed" connection which means that when there is no voltage applied to the coil of the relay (or control pin of a solid state relay), the voltage through the relay will be switched on. When you apply a voltage to the coil then the switch will be opened and the voltage through the relay will be turned off.
The way it would work is to connect the switched +12V to the switched input pin of the relay (usually marked Pin 30). Connect the +12V lead from your handle bar lights to the normally closed pin (usually marked Pin 87A). If you don't connect anything else, your handle bar lights should come on when you turn the key on because the relay switch is normally closed (ON).
Next connect a tap from the turn signal +12 to the relay coil (control) pin (usually Pin 86). Then connect the other side of the coil to the system return (ground). This can be the same ground wire as for the turn signal and the handlebar lights.
Now, when you turn the key on, the handle bar lights will come on. When you turn on the turn signals the power to the turn signal will activate the coil in the relay and open the switch turning the handle bar lights off. It should be noted that this will cause the handle bar lights and the turn signal lights to flash in opposition to each other: When the Turn Signal lights are on, the Handle Bar lights will be off and vice versa.
Did I just make a simple mod way too complex? Hope not. I includes a simple schematic for illustration.