I apologize if I’m way off base, but this is what I understand from quickly looking at the wiring diagrams. I also checked on my car and the fender indicators only flash with the turn signals/hazards and do not come on at all with the headlights. Looking at the wiring diagrams I suspect yours to act the same, however my car is not US Spec.
The headlight system is ground switched meaning that 12V always sits on the circuit/bulbs and a ground is made at the switch to complete the circuit. The other lights (as Mike states below) are the opposite and switch 12V power with a constant ground. If everything is working fine until you turn on the headlights and then your indicators light up, it sounds like you have a short between your headlight circuit and the indicator circuit. The side markers lighting up when your headlights are on is indication that your headlight circuit is providing 12V to your indicator circuit (aka short between the circuits) which in turn utilizes that circuits built in ground. The turn signals won’t work then because that circuit already has the 12V that the flasher would be providing and the lights are already on.
I see you said you were doing a resto. I am not sure what all you have touched but at first narrow the scope of your inspection to the areas you touched. If you didn’t remove your dash or touch wiring in it do not start by ripping away at things in there looking for the problem. Only after elimination of the areas you worked in should you dig deeper. In regards to areas you were working in, at first I would inspect connectors as these are the most handled items so they are more susceptible to damage from flexing, tugging and such. Next look for exterior damage on your wiring bundle (chafing, pinch points, etc). If you find any peel back the tape/wrap and check for damage to the wires’ insulation. Also, if there are any crimps in the system look those over for any wires that may have escaped them. Then slowly widen your scope as you eliminate areas.
Past visible inspection you’ll need some sort of device to test with, like a multimeter.