Here's a video that may help give you a little more confidence.


The demonstration starts at 2:55.

There's another video somewhere showing and empty bike rolling all the way down a hill. It flopped over only when it lost forward momentum.

In another incidence, while I was mowing the lawn one day, I say a teenager riding a bicycle that was too small for him. He was riding no-hands and pedaling really slowly. So he was moving along really slowly while he was texting. Every time the bicycle looked like it was about to flop over one side, the front wheel naturally turned to that same side and the bike would roll in that direction instead of falling.

If you can keep your body on the bike while it's moving, you won't fall over. It's your nerves that's making you fall. See if you can get on a stationary bike at the gym and just sit there without holding the handlebars.