A page can only show things below the browser's tabs and address bar, which keeps the teleprompter further from your camera than it could be. In Chrome and Edge, the new Pop out button moves the teleprompter into a small window of its own that stays above every other window. Drag it to the top of the screen, beside the camera, and resize it by dragging its edges, so you look at the lens while you read.
The script, scrolling, speed, reloading and the space bar all work in the popped-out window. Closing it puts the teleprompter back on the page. If you share your whole screen, the window appears in the recording, as the teleprompter does now.