I skip white shorts now.
I used to buy Nike and Wilsons because they have the best shorts. Very thin and easy to move in. Then I get the white versions of the same model in nike, yeah they are see-through but still thin.
I buy white Wilson's that are the same model as the colored ones and they are thicker! So I asked Wilson and they mentioned that all of their white shorts have an extra layer to prevent the see-throughness.
So to avoid the situation I stick to colored shorts.
However if you need white and don't want the transparency at least we know that Wilson is always double layered.