I've had much more luck at thrift stores with business/business casual wear and some jeans than I have had with t-shirts. At this point, I really don't bother looking there. Going up market may help some, and paying attention to the weight of the material will also help. Of late, the retailers you mentioned among others have been selling 'tissue' tees that are probably half the weight of a normal tee. It's kind of remarkable marketing if you think about it; they use a much cheaper, lighter weight material, and you have to both buy two tees to wear at the same time and replace them twice as fast because they get holes in them. (That probably irritates me the most. My clothes as a kid never got holes. I should not, as a grown-ass-woman, be putting more holes in my tees now than I did then.)
Anyways... at least at Target, you may have better luck with their tees that are marketed more towards women than teens. Their Merona line seems to be a little bit heavier. I've also been pretty pleased with Lands End's quality and sale prices and they've added cuts that are more trendy. I'm sitting here, in fact, in a pair of skinny cords I got from them three years ago that get fairly regular wear throughout our admittedly short winters. Even the inner upper thigh where my legs rub isn't showing very much wear. Also, don't bother with the Gap. I used to work there and across the year that I did (2006), there was a fairly marked decline. I don't know what happened, but things I got at the beginning of that year lasted years, almost nothing I've purchased since then has come close.
In other words, hun, it's not just you :-)