How can it be the code, when it's yer bitmap images that don't look curved?

Dreamweaver is a web page / site building programme.
The square image you posted is made up of bitmap images (Top, bottom, sides and corners).
It's the corner parts of these that you want to curve smoothly.

"Paint" isn't the world's greatest image creation application.
This may be the problem (I don't use it so I'm not sure on this).