St2 Indicator Wiring Defeating Me

The short answer is it looks like you may have a fault on the earth (negative) side of things and two cables have chaffed and are now making contact where they shouldn't, or within the handlebar switch itself. In the first instance, and taking into consideration what you've done so far, I would locate the main loom connector - it's a big chunky block connector where most connections pass through and certainly your indicator circuit ones do, pull it apart and give it a liberal application of electrical contact cleaner spray. Check it visibly, re-connect and try them again. I would also recheck the connections in the handlebar switch itself as for both LH and RH to light up implies there's a faulty connection within the handlebar switch if it's not a chaffed wire somewhere else in the circuit.

The long answer to your question is as follows. (NB. I There are others eg @Derek and @Chris who are more experienced than me and can/will correct anything I have misstated.) Please reference the attached wiring diagram.

When you turn the key on power goes to the LH switch assembly via the orange wire.
That feeds the blue wire which gives power to the indicator flasher unit.
The black wire coming out of the flasher unit is a common earth that links back to the flasher unit from all 4 indicators and is connected back to the battery.
The pink wire that comes out of the flasher relay feeds power back to the LH switch assembly.
If you look at the LH switch diagram the turn switch is in the middle, the triangle shaped affair, which shows the 3 positions (confusingly all linked up). Picture a toggle that hangs from the top pf that triangle and is permanently connected to the pink wire.
That toggle is live but when hanging straight down it's not connected to anything so the indicator circuit is not complete.
When you ask for the LH indicator you move that toggle across to the right in the diagram which connects the pink and the green-white wire. That feeds power to the left rear indicator, completes the circuit to earth (the black wire) and the rear indicator comes on.
As the front indicators are in effect permanently wired on, as soon as the earth circuit is energised at the rear indicator it causes the front LH indicator to light up and the dash indicator which is connected to the front indicator by a green wire also lights up.
The flashing is caused by the flasher unit breaking and re-connecting the circuit.
When you cancel the turn signal the toggle moves back to the broke circuit position in the LH switch assembly.
When you indicate right the same thing happens again, except this time the toggle moves the connection to the left in the LH handlebar switch (as depicted on the diagram) and energises the black & white cable to the rear right indicator which intern completes the earth circuit and the front RH indicator and dash indicator light up as well.

I bet you wish you'd never asked now

In your situation the LH circuit is fine. The fault lies in the right hand circuit which is also energising the LH circuit when you want to turn right. As the breaker for that circuit lies within the handlebar assembly itself then I'd double check that part of the circuit is functioning correctly. You may find the toggle switch is making contact with the green/white wire terminal as well as the black/white wire terminal when it should only be making contact with the latter.


NB the reason the dash bulb is so faint is that there probably isn't enough electricity flowing to light up all 4 indicators and the dash, which being the lowest wattage bulb in the circuit makes the least demand upon the available electricity.

 

Here’s a simplified indicator wiring diagram.


If all four indicators are flashing then there is a fault with the switch or with the wiring. If you can lay your hands on a 3 way switch (the sort used on electric Windows). You can use this to operate the indicators. If they work then the switch is faulty. If not, then I would suspect the wiring loom. Check for visible chaffing or breaks. A multimeter will help in checking the supply cables for continuity.

 

@chrisw That circuit isn't quite right for many bikes including the ST2 and most Ducatis. The warning light is wired across the left and right sides so that when one side is on it also provides power to the warning light which finds its earth through the bulbs of the other side. Because the warning light is a very low wattage it doesn't pass enough current to light the bulbs on the opposite side.
However, as some folk have found it, if you fit LED indicators it will provide enough current to light the opposite side and you end up with all 4 indicators blinking together like hazards.

Here is a more representative circuit for the indicators on the ST2.

 

If you want to use LEDs in the indicators and the telltale you need a complete electrical separation of the left and right hand sides and a flasher that is LED compatible.