Hi yamaha_pilot,
There are a few reasons why you could have the symptoms you are.
One of the first reasons would be if an element in one of the turn signals on that side is burned out. The front turn signal has a running light element and turn signal element. You should turn on the right side and check to see if the turn signal element is actually coming on front and rear. If it is not, you need to check to make sure you have voltage at the socket. If you do, and the bulb is not coming on, then more than likely the bulb has burned out. The other option would be if the socket is not grounded. If you have a known good bulb and positive voltage to the socket and the light does not come on with the turn signal switch, you will need to investigate the ground for the turn signal that is not working.
The symptom of the odometer, electrical panel and etc. going dim could be an indicator of excessive amp draw, but the probability of that happening without blowing a fuse is remote.
The other question I have would be if you have attempted to install after market turn signals. If you have, then I would try reversing the two wires you connected for the turn signal.
For the record, left turn signal wires are chocolate (brown), right side are green, and running lights are blue.
If you have any more questions, feel free to ask.
Thanks,
RSRBOB

Hi yamaha_pilot and thank you for the additional information.
First, no, the flasher is not a possibility. Both left and right turn signals use the same flasher. Therefore, if it were the flasher, neither side would flash.
One thing I am not clear on is whether you have disconnected any of the stock wiring. I believe when you said you "have" the stock turn signals, you mean they are still on the bike. A double black plug is just a double plug for ground. You should not having any other colored wires than black plugging into that plug. Did the right side turn signal just quit working one day, or had someone been working on the motorcycle, and then after they got done they didn't work anymore?
I do not have a simplified wiring diagram, but it really is as easy as I described it. Since you said you found a green wire, you are looking at the wire for the right side turn signal. The front turn signals will have 3 wires coming out of the turn signal, Black is ground, Blue is running light, and green is for right turn, and chocolate would be left turn. You should find the corresponding colored wires in the immediate vicinity of where the turn signal wires stretch to. The rear turn signals are the same, minus a blue wire. So you would only have 2 wires coming out of the each rear turn signal. I suspect since you mentioned the double black plug, you are looking at the rear turn signals. If that is the case, and you seem to have an abundance of black wires, remember that the tail light will have black ground wires that need to connect to the harness as well.
Hopefully that clears it up for you. If not, answer my additional questions and we will try again. Also, if it didn't work, tell me if you see the front bulb in the turn signal get brighter when you turn the turn signal switch on to the right, or does the back even come on at all with the turn signal switch?
I still suspect you have a green wire plugged into a black connector or something.
Thanks
RSRBOB

ok, let me know.