So you're missing one function, if you assume you have 3 LED functions trying to do 4 bulb functions, and can't do split-voltage control because they're meant to be trailer lights (everything is turned on with 12VDC, instead of different voltages for different functions).
There are trailer lights that have the STOP and TURN combined on the same wire, and lights that have the STOP and TURN separated. That is determined by the type of trailer connection, because you need more wires/signals for one than the other, so need extra connections in the trailer connector, etc. The combined functions would usually be handled by the tow vehicle, like the Trailer Tow Relays that are called out on that schematic.
You seem to have trailer lights that have the STOP and TURN combined, if my intuition is correct, if you only have 3 controlled wires. You might be able to tap into the trailer connector wiring and use the combined STOP/TURN function?
I see from the wire nut picture that you've wired:
Brown/LtGrn for the backup
Blue/Red that seems to be license/park
The Tan should be turn signal, but what other wire have you connected to that (White/Tan?).....?
If the turn signal is Tan, that additional wire is probably tying both lights together such that they both flash at once....? Presumably you're tying the brake and turn signal together.
EDIT: I see that the White/Tan is to the trailer tow relays, but the rest of the trailer tow relay info is missing from that schematic so can't tell if there's anything wrong there. I would almost guarantee that's the issue though, as you're likely bridging the two separate turn signals together with the White/Tan.
That's easy to test though, just remove the wirenuts with the TAN and White/Tan on BOTH the LEFT and RIGHT, and check continuity on the White/Tan between LEFT and RIGHT sides. If they're electrically connected, that's your issue.
You would then need to find the proper outputs from the trailer wiring for the combined TURN and STOP functions (probably NOT the Tan wires), if you still want to use those LED lights.
In other words, it would appear that you're grabbing the Tan wire from the input of the trailer tow relays,, and you would likely need to use whatever wires are for TURN/STOP on the output of the trailer tow relays.
Everything you need should be there somewhere, you just need to find out what it is. You've got a trailer connection, and you're using trailer lights, sooooo.......