1. Photocopy the piece and tape the sheets together. Although the copyright laws require that you play copyrighted music from the original scores, no one will blame you if you photocopy the piece for your own convenience but have the originals with you.
2. Go to printing and photocopying service and ask them to cut and bind the collection (my favorite). They will cut the borders and use spiral coil binders. This way your pages will easily open 360 degrees.
3. Adjust the music rack so that it is more horizontal. The problem with flipping over pages might also be the position of the music rack on your organ. If it is too vertical, some books will not stay open. If you could adjust the music rack to the more horizontal position, then often it will be enough to keep the pages open.
4. Find a different edition of the piece in horizontal page layout. Books with horizontal page layouts will stay open more easily than the vertical ones. Because there is so much variety of organ music editions, often you can play from such a collection. However, many modern organ composers have their music published with only one publisher so your choices might be limited with modern music.
5. Play from tablet PC. As the technology is getting more advanced by the day, many performers nowadays are choosing to play not from the printed score but from tablets. You can either find copyright free music online or scan your originals to use with your tablet. You can even find a device (it looks like a pedal) with an app which help turning pages without the use of your hands.
6. Play the piece from memory. If you could memorize your piece, then you won't even need all the previous tips. Of course, memorizing takes time and you should follow a systematic approach, if you want to feel secure in your public performance without music.
Apply some of these tips when you prepare to play your pieces in public. They will give you some piece of mind and you will feel much more confident and secure during the performance. After all, you have so much to think about besides keeping the pages from flipping over so it's better to take care of this problem ahead of time.
By the way, do you want to learn my special powerful techniques which help me to master any piece of organ music up to 10 times faster? If so, download my video Organ Practice Guide.