I was only a toddler when me and my family went to ride on the ponies at Griffith Park. The ponies have special saddles/harnesses to accommodate small children so they won’t fall off. And if I recollect correctly, you could pick if you want a slower or faster horse too. Once you’re strapped in securely, you wait in line till it’s your turn to make your way around the path (it was just a circle). I remember when the horses were side by side, my legs would get pinned between the two horses. It didn’t hurt, but at that age it had me a little concerned being smashed between two huge animals.