Left side turn signal not working properly

Has the vehicle been in an accident on the left rear side. First check the bulb to be sure it is not defective, trade it out with the side that works properly to test. If the bulb works well then My thought are you may have a shorted wire. The turn signal flasher works by heat build up within the flasher the heat is caused by the load of the bulbs, the load heats the metal spring causing it to open the circuit then as quickly as it cools down the circuit closes again lighting the bulbs.
There is also a possibility of a defect in the turn signal mechanism on the steering column. The turn signal switch on many models does three different things on many models that use the same bulb element / or bulb for brake and turn on rear of vehicle. It controls the circuit for brake light by sending power both left and right side rear when turn signal is in neutral position then when turn signal is in left turn the switch cancels brake light power to left side rear and supplies flasher power to left side while maintaining brake light supply to the right side rear, and so on opposite when the right turn is chosen on turn signal switch. The turn signal indicators on the dash normally run off the same circuit as front turn signal bulb only which is also controlled by turn signal switch. Front has turn signals but no brake light.
I think there is an additional load on the left side rear turn circuit and it could be either a wire that is shorted or the turn signal switch if the same bulb element is used for both functions.
Often times if trailer lighting is connected the flasher will also blink a little faster because of the extra load of additional bulbs.

it's the grounds if they're both flashing fast it's definitely grounds take a test light and see if its got power goes through the test light grounded to the frame and if the light comes on then it's definitely your grounds

The 2003 Softail TSM/TSSM's turn signal feature provides for Automatic cancellation; Manual cancellation; Four-way flashing and Diagnostics mode.

Turn signals can only be activated or deactivated with the ignition key in the "IGN" position, not the "ACC" position.

If you press the left or right turn switch the signal will activate both the turn signal operation/illumination and the automatic cancellation function. If you press the button a second time the signal will cancel. Also if you have one signal on and you press the opposite signal switch button the first set signal will cancel and the second pressed side will operate. However if you do not press the signal switch button the second time the signal will remain on and will not cancel while you are approaching the turn but will only automatically cancel after the TSM/TSSM senses that the turn has been completed because when the signal switch button is depressed the TSM/TSSM starts to count to twenty after which it will cancel the signal function if speed is not reduced below 7 MPH but as long as the speed of the motorcycle is above 7 MPH or is stopped the signals will not cancel until speed is resumed to above 8 MPH at which time the 20 count will resume and the signal will cancel after the end of the 20 count. The signals will also cancel within two seconds after the bank angle sensor in the TSM/TSSM senses more than a 45 degree lean of a turn action after which the motorcycle returns to a normal upright riding position. As well the TSM/TSSN will calibrate itself after it senses that he motorcycle has been ridden for one quarter of a mile or four tenths of a kilometer at steady speeds in an upright position. ON

The TSM/TSSM also flashes the signal simultaneously as four way flashers whenever the ignition is set "ON" , the security system is not activated and both signal switch buttons are simultaneously depressed. If you have to leave the motorcycle where it is with the four signal lights flashing they will continue to flash for two hours if while they are flashing you turn off the ignition switch and activate/arm the security system but if you turn the ignition switch is turned on, disarm/deactivate the security system and depress both signal light switch buttons simultaneously the four lights will cease to flash before the two hour period is up.

An auto arming feature that is not always active on domestic motorcycles but is user selectable on these models but that is always active on HDI models Auto-arming requires that the four-way flashers be activated within 30 seconds of key OFF or after the vehicle has been disarmed. This allows a vehicle to be left in a secured state with the four lights flashing until they are turned off as described above.

If the four way flashers are not turned of before the security system is disarmed/deactivated then the TSSM will stop four way flashing for one second and then per flash the lights one time to signal that the system has been disarmed/deactivated and then after one more second the four way flashing will resume.

When there is a burned out light bulb on one side of the motorcycle, the remaining light on that side and the turn signal indicator light on that side will flash at double the normal rate after the fifth flash so if that happens check your signal light bulbs.

As the TSM/TSSM also diagnoses for signal wiring short circuits and opens AND for stuck signal light switches if there is a stuck turn signal light switch the automatic turn signal cancellation feature will not work and the signal will continue to flash AND if there is a stuck turn signal switch the four way flashers will only be activated if both of the turn signal switch buttons are held in for more than one second so if the four way flashers are on when a signal switch becomes stuck you may have to hold them in for a longer period of time to cancel them also.

Otherwise if you are having turn signal or four way flasher problems you should check for TSM/TSSM trouble codes.

Have you tried to deactivate them?
To Activate
Turn the ignition switch to IGNITION with security system disarmed (fob present, if equipped).
Press the left and right turn signal switches at the same time.
Turn the ignition switch OFF and lock the switch. The four-way flashers will operate for two hours. The security system (if equipped) will automatically arm.
To Deactivate
Have the fob present to disarm the security system (if equipped).
Unlock the ignition switch and turn the switch to IGNITION.
Press the left and right turn signal switches at the same time.

The switch contacts are stuck. It might work if you disassemble & clean it, or you need a new one.

Has the key been damaged? You should make sure that the vehicle is in properly engaged in park if it's an automatic.

Gday mate, we had a problem with another late model vehicle similar to that, the bulb that was put into the reverse light was not exact brand product ( ohmage was slightly wrong ) and fowled up computer, correct bulb in place and disconnect battery to reset K A M ( keep alive memory) problem solved, BEWARE, some vehicles cannot have KAM reset except for dealer computer tethered reboot , just try putting the factory correct bulb(s) in and see how it goes, geoff. hope this helped.

although you can cancel the left turn signal there you could still have a faulty switch which is not turning on the right side signals

also have you checked to see if the bulbs are bad, as sometimes a short can blow both front and rear bulbs

first just because a bulb looks good doesnt mean it is couple of peices of wire and a battery is all you need to test the bulbs secound thing the turn signal cancel works on the amount of flashes regardless of which side is on note:any bulb thats bad none on that side will work

Turn signals flashing is the confirmation of alarm system being on.I thought both sides flashed but if alarm is faulty guess that's messed up too. Try starting from stratch with alarm,disarm it[disconnect neg wire at battery]then reconnect wire and when that sets off alarm turn it off in normal way.That should get things back on track.Plus,try replacing the flasher unit,don't know why this works but I've heard of it.