This is a great question and I understand your concerns.
First, xrays are very important. They show us cavities between the teeth and if any adult teeth are not forming. The dentist’s primary job is to catch little problems early -and solve them! – before they are big problems.
The radiation concern bothers us too. The good news is that new technology allows xrays to have very little radiation. We also protect little people with lead aprons. In Phoenix, your child is getting more radiation playing ouside in the sunshine than from xrays. However, radiation is cumulative and you should inform your dentist if the child has had more than one MRI or any radiation treatment.
Now, back to your real question: I would suggest taking xrays on a 4 year old if the 4 year old will allow it. Cavities that can be seen without xrays are generally very large. Xrays allow us to see the little ones. Two xrays could mean your daughter gets a small filling now instead of a child root canal and crown later.