we have car accidents now but i was wondering?

Not just in the west: horses were used all over the country. Other countries, too.

It was actually very easy to flip over a stagecoach or wagon. A wagon or coach has a high center of gravity, and a crummy suspension. Roads were poor, too. There weren't that many wagon-to-wagon crashes, but mostly because traffic was light.

Horses themselves tend to not crash into each other. They don't drink beer or whiskey, even if their riders did.

Human stupidity Trumps *anything*

Horse Sense is why we did not have many vehicle accidents until we started cars.