You basically need to just go through the WebBrowser.DocumentText looking for a <link tag, and if has rel="icon" or rel="shortcut icon" before the next > (the closing tag) then it is a favicon link, and you can search for the href=" part of the tag and download the image using the webclient.

And I'm not really a heavy GDI user, so I'm not entirely sure how you can draw the image onto the textbox. I see now that it doesn't have any image properties like a label does, so you will have to do that. Try searching the forums for GDI programming, I'm sure they're something you can use for that portion. But first you need to get the image downloaded. You might also want to just use an ImageCombo if you are using this for a browser address bar. Then it would be easy to add the image to the current item.