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I have used Traffic++V2 and it doesn't work the way I need. Arrows on lanes changed but cars still went where they were not supposed to go.
I guessed what it is for cars only already started their travel.
But even after hour of playing I can see cars which still turn left even if I forbid this.
Also all 3 lines of highway was opened for one and only way to turn right but whole traffic still stay on 1 right lane. :(
 
I have used Traffic++V2 and it doesn't work the way I need. Arrows on lanes changed but cars still went where they were not supposed to go.
I guessed what it is for cars only already started their travel.
But even after hour of playing I can see cars which still turn left even if I forbid this.
Also all 3 lines of highway was opened for one and only way to turn right but whole traffic still stay on 1 right lane. :(
I know when I started using Traffic++ it took a while for cars to start obeying the rule. But I use it a lot, exactly for this purpose, because it speeds up traffic a tonne, and for me it works beautifully.

Not had any experience with Traffic President, so can't help there, sorry.
 
If you want to be 100% sure, clear all traffic, after that newly spawned cars will obey to road rules (in this case won't take wrong turn).
The reason why your cars still take forbidden turns is, that cars have set their route before you have applied changes to intersection. Why it took so much time and cars stil do that? because these were either stuck in traffic or spent a lot of time somwhere in the city.