For public bathrooms, my daughter was a bit scared, especially of noisy or smelly ones, and didn't like to put her hands on the seat to hold herself up.
After struggling with one of those dang folding public toilet seats for a couple weeks, necessity became the mother of the hack when I forgot to bring it. So here's what I do: seat myself (clothed of course) on the solid back part of the seat with my legs straddled. This "shortens" the opening just enough to make it comfortable for a little one, and allows me to wrap my arms around to help with the scary part.
Works better, and less to carry.
To be clear: she acts as a warm, huggable public toilet seat for her kid.
Charisse: you deserve a medal. We all do, don't you agree?
More: Hacks for public restrooms