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That does NOT look like a wart. It looks like a tumor or an abscess. If it's a tumor it'd have to be surgically removed, and if it's an abscess it might have to be drained. If it's growing that quickly, I'd guess it's an abscess.
In some cases, even if it is a bacterial abscess, the antibiotics won't be able to reach it, because the body has physically blocked it off. Also, it would have to be the right kind of antibiotic for the bacteria. Make sure that your vet does a "culture and sensitivity" test next time your bird is on antibiotics, to make sure he's going to get a medication that actually works.
What were the antibiotics for? What kind of infection does the vet think he has? Any symptoms?