If you create right turning (dedicated) lanes - with either TM or T++ - it appears that the commercial buildings behind the intersection will not receive goods. What's worse it'll cause random commercial abandonment throughout the city (I'm not sure exactly why). It's not distance or traffic jams - the businesses simply stop ordering goods.
Took me 2 cities and several weeks to see why certain commercial buildings kept running out of goods. They were all placed on 6 lane high traffic roads. To alleviate the traffic I usually split the intersection as to create dedicated left, forward and right turning lanes. Just to clarify - while the traffic on the roads was high it was free flowing - absolutely no jams or even slowdowns.
I finally cracked the issue on a new town as commercial began massively abandoning after I applied the intersection lane split. I've then redone the intersection as to only have a dedicated left turning lane - ie. lanes: left, forward, forward/right and the commercials came right back.
I had both TM and T++ installed (and so used the T++ lane changer) on the latest city where I confirmed the issue. The same thing happened a while back on a city I only had Traffic Manager installed and used it's own lane changer to do the same split.
https://forum.paradoxplaza.com/forum/index.php?threads/psa-not-enough-goods-to-sell-tied-to-traffic-changing-mods-traffic-and-traffic-manager.859104/