Starting approximately 12/13/2013, my Kindle Fire HD started behaving strangely when scrolling through web pages in its Silk web browser. The part of the page would get stuck and the new part would "accordion" over it, like a tile. It is difficult to explain, but very annoying.
I suspected a hardware problem or a software one. I was considering resetting the device and, failing that, getting a new one, but did some online searching and discovered that this issue started to happen about the same time for others using various Kindle Fire devices (HD, HDX, 7 and 8.9 inch size).
It appears that a recent software update is the culprit and Amazon is working on solving this issue. In the meantime, you can enable direct software installation and download/install Chrome or Netscape Navigator.
Update 12/26/13: my Kindle Fire HD fixed itself. Apparently a software update resolved the issue that was introduced by a previous automatic software update. Nice.