Attempts to explain a complicated thing always seem to come off like this...

I just started working at a bed and breakfast and was finally taught in person how to do this properly. A large table surface is helpful.

1. Put each hand into two of the corners that are closer together (the width, not length of bedsheet).
2. With hands still inside, insert both corners into the opposite corners of the sheet, so you are folding the sheet in half (hamburger style, not hotdog style.
3. Now that all corners are accounted for, put your hands together and fold one side over the other. After some beating and shaking, you CAN have something that looks close to picture #4 above, it wont be perfect, but it doesn't matter.
4. Take the less-perfect side and fold it down a third, then fold the other third over that. This will make it look pretty close to a good rectangle.
5. From there, fold into fourths or thirds, and the sheet should be pretty perfect.

All that said, it's much easier if someone shows you in person.

Edit: Also, beating and shaking are pretty essential for every step. The thing just needs to be fiddled with.