I went to the local bike shop and got my 10 year-old daughter a small used Trek mountain bike. It must've been their smallest model with the 26" wheels. It's a quality bike, tuned up by the shop. Right now, she's 4'11 and the seat is all the way down. I think I spent $150; a lot compared to a Walmart bike, but not much compared to a new Trek!

It's just a bit big, but she can handle it and it will work for several years. She's 11 now and it's fine. My 17 year old (5'5) still rides her bike with the same size frame; she's not big on full leg extension anyhow; likes the seat pretty low so she can touch both toes down at stops.

Before this, I bought my daughter a Trek BMX type bike, but she outgrew it FAST and it was wasted money.

My 11 year-old daughter now prefers to ride my folding electric bike. It adjusts for a HUGE range of rider sizes. With the seat and handlebars all the way down, it's fine for her. With them up quite a bit further, it's fine for me.