This is a Twitter thing, not a Magento issue. Here’s something you can try. Go to Twitter and click on “widgets” in the right side lower menu list (I use the new Twitter interface, so yours might be in a different place). Follow the instructions to create a new “widget” for “My Website” and then select to create a “Profile Widget.” Customize the look and feel here, then click “Test Settings” at the bottom. If everything works how you like it (you’ll see the actual widget on this screen), then “Finish and grab code.” Copy that code.
Now go to Magento admin and go to Static blocks in the CMS menu. There should be a “Twitter” block there. Open that or create a new block and then make sure to hide your editor, and then paste your Twitter code over any other code that’s there. Save it, and you should be good to go!