Turn signals not working, blowing fuse
My measurement with the multimeter is 0.03.
0.03 Ohms? That's pretty much zero. Most meters struggle with any low ohm measurements. Put the leads together and and might show the same. Doesnt really matter for what your doing but something to be aware of. For a comparison, my multimeter showed 0.6 ohms and 2.2 ohms for the readings across an 1157 bulb.
WHen you have time, start disconnecting and see if you can narrow down what component or section of wire is grounding.
If the ground is eliminated by disconnecting something, then check the something.
Earlier, I suggested starting at the flasher unit. I still think thats fine.
Keep in mind that if its a bare wire, then moving wires around may temporarily break the contact with the sheet metal.
That may have been what happened before.