#1. Light trucks don't cause sway.
#2. Trucks don't cause sway at all.
#3. Truck tires don't cause sway.
Trailers that are not set up properly cause sway.
Get the trailer set up correctly and the tail will follow the dog. Get it wrong and it won't matter if you tow a 20' TT with a Kenworth, you will get sway and you will feel it in the truck.
The problem is a lot of trailers are designed incorrectly now days to keep the tongue weight down. The manufactures are doing this so the average SUV owner or light truck owner can tow more trailers on the market. By doing this they are causing a lot more sway problems to the general public.
The bottom line is, get the trailer set up right, keep under the manufacture limits and you won't have any problems.