Side seams do not provide better rigidity to the shirts nor does it prevent shrinkage. The difference between shirts with sideseams and the ones without (tubular knit) comes down to style and cost. Tubular knitting also allows the designer/buyer with minimum fabric waste when designing the t-shirts. Tubular knits can cost a little bit more than shirts with sideseams which ultimately comes down to preference. Doesn't it look pretty cool to have no seams on a shirt?
Shrinkage and twisting is caused by the releasing of tension (torque) in the yarn after it reverts back to its natural state. When a yarn is spun, the cotton fibers are strechted. When you wash or steam press a shirt, the water molecues reacts with the fibers from the cotton allowing it to contract to a shorter state.
Have you ever twisted the outter wrapping of a straw until it becomes very stiff? What happens when you add a drop of liquid (water or soda) onto it? It unravels itself back to it's original state right? Similiar concept...multiply that with thousands of cotton fiber.
A lot of the shirts we provide are pre-shrunk - meaning they are prewashed, dryed, then rolled, before they are made into t-shirts. This minimizes the shrinkage during the first wash (which is where you'll see a majority of your shrinkage).
Hope this helps.