L.M.
My 4 year old really likes wearing boxer briefs. They came in size 2T and up in the toddler section, and XS in boys.
If you've already tried that, how about swimming trunks? They are probably on sale now, close to the end of the summer, and they have lining in them, which is almost like wearing underwear. My son has a few pairs that he wears as regular shorts because he thinks they are really comfy and loves the designs on them.