Mac-Welcome to the wonderful world of machine embroidery (ME)! You will be addicted to a terrific craft with lots of helpful, friendly people. Dumb questions answered immediately, stupid questions do take a little longer (LOL).
I do have a Blog on stabilizers, but it does not specifically address this question. So, I will attempt to do so here.
You don't mention which mesh stabilizer you are using, the one I have is very soft. So, I am not sure why you are having this issue. One cause could be a very dense design. T-shirts are, by their nature, soft and having a firm design would causing some stiffness. I know that some ME crafters have placed a backing of soft fabric on their completed designs for this reason.
Again, because of the nature of t-shirt material, it is prudent to sew on a modest design. A too dense/large design can cause the material to sag, not a pretty sight! I love to do 'outline' styles on t-shirts. I think it adds to the 'casual' look of the t-shirt.
Another thought is that you should be using a 'bobbin thread.' Bobbin thread is somewhat lighter and therefore less dense. Overall, I do recommend prewound bobbins and that will help your issue and make your work easier and much more professional in its finished product. Relative to your time and effort to wind a bobbin, they are a very good buy. It is the "right tool" for the job. Ann's prewound bobbins are available in both black and white and very reasonably priced. You get 144 bobbins, enough to keep you busy for a week or two.
Pat