Norazdad, Depending on the year of your bike the turn signals are pretty straight forward. The early years power from the key switch went to the flasher on a brown wire. From the flasher to the left/right switch, from the l/r switch to each side.
Dark green to the right side, Chocolate to the left.
Some years used two indicators on the dash, some just one. The two indicators were just wired one to each side. On the single indicator it was wired to both sides.
When the right side was flashing the indicator grounded through the left side, the current draw through the opposite bulb was enough to prevent the opposite side from flashing.
On the 78 they added the self canceling system. I'm not exactly sure how it works. The basic system is the same with a few changes. The turn switch, the lever self centers after you push it. There are several extra wires involved. The third wire on the flasher was changed to green/yellow, the switch has the brown/white. dark green and chocolate as well as a red/yellow and black. They added a reed switch in the speedo, as well as the self canceler unit.
My best guess on how it all functions is that when you push the turn button, power gets sent to the self canceler. This tells the canceler which lights to light and trips a latching relay. This starts a clock and reading the distance traveled by means of the reed switch in the speedo.
After a combination of time and distance traveled the self canceler grounds the circuit unlatching the relay, stopping the flashing. Or if you want to stop them earlier you can just push the button straight in.
Now onto your issue. If your turns light up but don't flash. This is most often a low voltage condition, as in not flashing when the engine is not running or at idle. This is sometimes corrected by cleaning all the connections in the circuits. Some times it can be your charging system isn't up to the task.
With all stock parts if they don't light it can often be a bad bulb or weak connections at the fuse. The stock round glass were crap when new and 30+ years haven't helped.
Swapping the stock fuses for modern blade type fuses is a very good upgrade.
One way to check this is check voltage at the flasher on the brown wire. It should be very close to battery voltage. I start these checks by turning on the key, then check voltage at the battery, this gives you actual voltage thing get in the operating mode. Now use this voltage as a base voltage. compare all your test voltages to this one.
If all the connections between the battery and where you test are in good shape your reading should be within a .2 or .3 of battery voltage I might let them get to .5 low but no less.
If the voltage on the brown wire is ok then make a jumper wire to jump from the brown wire to the brown/white. This bypasses the flasher. Try the turns, do they light? If so the flasher may be the issue.
If you have swapped the lights for after market, the bulbs in them are often 23 watt bulbs. The stock flasher is load dependent. It needs two 27 watt bulbs and the three watt indicator to work.
Check the bulbs them selves. If they are 23 watt bulbs they won't work with the stock flasher.
Easy fix, swap the stock three prong for a parts store two prong flasher. I can't recall if it's 552 or 522 but it's a good flasher and isn't load sensitive.
If you replaced the stock signals with LED then you will need an LED compatible flasher. I like the LF1-S-flat from
www.superbrightleds.com It comes with the leads going to a plastic block. The leads can be removed from the plastic. Plug the red lead into the brown wire, the black into the brown/white wire of the stock flasher socket.
On the early bikes this is just fine, no real changes. On the later self canceling bikes this deletes the self canceler. No real problem, just have to learn to turn them off yourself.
Sometimes on the signal lights them selves the grounds can get weak. They ground through the body of the light. Where the mount bolt comes through, both front and rear there is a black wire with a large ring connector, it goes over the mount bolt under the nut. The other end of this wire has a bullet connector that plugs into the harness ground. A multiple female connector, one in the headlight bucket, one under the seat.
Clean bare metal to clean bare metal is a good ground.
Hope this helps.
Leo