Turn signals only work when fuse is out?

I found this in the shop manual: i1268.photobucket.com/albums/jj566/Tom_M..._499_zpsjbzyfyaq.png

• It's not the battery
• The Bulbs are all the correct wattage (but shouldn't matter with new digital relay
• Relay is brand new

It must be a wire. I'll go through all of the turn signal related wiring.... uggh

In that mini diagram, the wire from the ignition switch to the turn-relay is the brown wire. But in reality it goes to many other things.
You said there is 0v on it, so if it's hooked up as shown in the full-size diagram, the horn would be dead too.

So it does sound like a wiring issue. There are ways to trace it out with a meter.