Hi and thank you for your question.
First, we need to make sure we have the correct color wires for the turn signals.
Yamaha uses Dark Green for the right turn and Chocolate (lighter brown) for the left turn. Yamaha doesn't do us any favors because those are not the only green wires or the only brown wires in the system. The one thing they do to try to help us is use Green for right because it has an R in it for right turn and Chocolate (not called brown) because it has an L in it for left turn. Blue is going to be for the running light. The red wire in the bike's harness is system power straight from the battery and runs from the battery to the key switch.
Now that we have established the correct colors for the turn signals on the bike's wiring harness, we need to determine/confirm what the 3 wires coming out of the aftermarket turn signals are supposed to do.
We know that one of the 3 wires is going to be ground, one is going to be for the turn signal element and the 3rd wire is for the running light.
If you don't have a wiring diagram for the turn signals, you can either contact the manufacturer and ask them or start testing the wires yourself.
I have seen the black wire being the ground wire on some aftermarket turn signals and sometimes the black and white wire in the turn signal is ground.
To test this for yourself, I suggest using jumper wires to connect the (-) negative terminal of the battery to the black wire of the turn signal and the (+) terminal of the battery to the Red wire on the turn signal. What we are looking for is the combination of wires on the turn signal that produce the brightest glow. Then leave the (+) to the red T Sig wire and switch the (-) to the black white wire in the turn signal. One of the 2 ways should be the bright glow of a turn signal. If you get a very dim glow, you do not have the ground wire on the TS. That might be the black wire. So if (+) to red and (-) to black and white produces the brightest glow, take the wire from the (+) and go to the black wire. This should leave you with the black wire and black and white wire connected to your battery. For simplicity I further suggest making your wiring diagram of the aftermarket turn signals so they don't get mixed up later.
Armed with the correct wiring of the turn signals, go to the bike and find the turn signal wires, ground wire, and running light wires.
I suspect you have blown some fuses because you connected a hot wire from the wiring harness to a ground through the turn signal. Typically a short would not cause a headlight bulb to blow. The "signal" fuse among other circuits protects the tail light circuit so I suggest checking that. The head light circuit should be on its own fuse, separate from the signal fuse. Also, I do not suggest trusting your eyes to tell if a fuse is blown or not. Use a test light or a volt meter and check both sides of the the fuse to make sure you have power in and power out. It happens often enough that a fuse does not look blown but it really is. By testing it electronically, you can eliminate that variable of overlooking a blown fuse.
One last question, are the new aftermarket turn signals LED turn signals? If so, you may end up with even more problems with the ignition and fuel management systems too.
Let me know where you get with this and we can go from there.