Have you changed the front bulbs? The parking light bulbs?
Check the turn signal fuse. (although I'm not hopeful on that).
If none of the above work....you've got a wiring problem. And that's a SOB to find. It could be a ground issue or something in the turn signal switch. Turn signal wiring is a black haze to me, I don't understand it very well.....no probably not at all.
There is a clue however. Let me repeat the problem
1. The turn signals work fine when the headlight switch is off.
2. You turn the headlights on and the turn signals no longer work.
Easy enough
The only difference is that pesky brown wire from the headlight switch. That wire doesn't enter into the picture unless the headlights are on. So, I would measure voltage at that brown wire. I think that could be where the problem lies.