You’re right to ask about the exposure to radiation with repeated X-rays during childhood. X-rays are the most accurate and reliable means of measuring limb length. But researchers at Cornell University are looking at the accuracy and reliability of other imaging methods.
They compared X-rays to CT scans and MRIs. The results from all three imaging techniques were checked against direct measurement of the bone with electronic calipers. CT scans and MRIs were just as good as X-rays. They are more expensive and require the child to hold still longer but there is no exposure to radiation.
Feel free to bring this concern up to your physician. Perhaps the X-rays could be taken less often (for example, every two years instead of every six months) with other means of imaging in between.