You will probably need to start spending more than Target / Old Navy prices, unfortunately. It is a tough transition for the wallet.
And measuring her would help, if she is comfortable with that (can get a rough estimate over her bralette even).
American Eagle / aerie often has cute suits that aren't too skimpy, so definitely try there.
J Crew & J Crew Factory *might* have something that will work.
And look into the more athletic and surfer-oriented brands, as they are generally designed to stay put when swimming, diving, playing volleyball, surfing, etc., rather than just look cute while lounging poolside. Quite a few brands have already been mentioned: Speedo (they do make bikinis), Athleta, Title 9 (much of what they carry is not house brand, so you might be able to find it somewhere else, too), REI, Carve, Roxy, Billabong....
Unless your teen is really good at pulling off ironic, retro looks, Lands' End is not likely to be a hit.