Caregivers generally keep track of how much radiation patients get, and even those limits are quite conservative/safe. And like others said, it’s very much one of those things where "a little is like standing out in the sun, a thousand times that is not."

Modern X-ray machines give much smaller doses (for the same type of scan) than older ones.

I have a retired radiologist in the family, and the lead aprons and other protective things were hard on their back (among other things). They're still dealing with it. And yet, family still worried about how much radiation they were exposed to. So… yeah, I have some respect for that job hazard.