Ah, there's your problem. The 5570 is a budget card.
Radeon HD cards use the following approximate naming scheme:
HD XYZZ, where X is the card generation, Y is the capabilities level (budget, good, excellent, premium), and ZZ is the individual iteration within the first two, starting with 50.
So HD 5850 is the newest generation, excellent-premium capabilities, first or second iteration of that card, while 5570 is a budget card which is in it later iteration.
EDIT: Similarly, nvidia cards of the latest generation tend to use:
GTZ XY0, where the X is the card generation, Y is the capabilities level, and Z indicates the level of additional goodies on the card (GT is barebones, GTS is moderate additions, GTX tends to mean more RAM and higher clock speed).
An HD 4850 or 5750, or a GeForce 250 or 450 should do just fine. If money is really not such a huge issue, a GTX 460 or an HD 5830 will put you comfortably over the point where you can get great FPS in WoW.