At the end of the first screen, it is possible that the screen exit sprite will not load in. If this happens, go left to off-screen it then go back to the right. This is due to having too many sprite on screen and no available slot for the screen exit sprite. See: sprite overload.
It is easier to jump over the red egg blocks just before the red switch at the end of the level, rather than going underneath them, because it scrolls the screen up, letting you see what you are doing better.