Here's my take on it.
Turn signal fix 12 Apr 2016.jpg
Only one side is shown for clarity.
Despite the apparent complexity it is really quite simple.
In the diagram the flasher switch is connected directly to the rear lights, the RC circuit, and to the normally open (NO) contact of the relay. When the turn signals are off the transistor is held in a non conductive state and the relay normally closed (NC) contacts apply battery voltage to the front light, keeping it on.
When the turn signals are activated the rear light flashes, and the capacitor charges up through the 10K resistor. This voltage is applied to the gate of the transistor which energizes the relay, connecting the front light to the flasher voltage. When the voltage from the flasher is interrupted the capacitor begins to discharge through the 330K resistor and the rear light, but before the voltage goes low enough to shut off the transistor the flasher voltage comes back on, keeping the relay energized.
When the turn signal is turned off the capacitor discharges until the voltage drops below the gate threshold, the transistor stops conducting, and the running light comes back on.
It sounds like you already know something about electronics, but if you need help with selecting components PM me for further information.
Oh, ignore the 500 ohm resistors connected to the LEDs. They are there only to satisfy my simulator program.