From what I can discern from question, you are actually looking for Video Mirroring, not Video out. They are two distinct features, using different API's.
Video Mirroring, i.e. everything that you see on your iPad screen, from home screen to various apps, is only available via Airplay, using an Apple TV, or using Apples Digital AV Adapter(HDMI) or VGA Adapter.
Video Out, which is what it sounds like you are currently using, is available with Apples Composite and Component AV Adaptors, and provides an output link for applications that are specifically outputting a video source, such as the Videos App, or YouTube. This would be why you can see YouTube Video's in YouTube, but not say when they are embedded in Safari.
You should also know, even Mirroring doesn't work for all Apps as you would expect. HBO Go for example, actively disables Video Mirroring and only supports Audio, so when you hook the iPad to your TV, all you see is and HBO Go Logo on your TV Screen. There is currently no official way around these restrictions.