Night-time dryness does not occur even up until 7 years old, and this is still normal, and also per our Pediatrician.
My daughter, was still wearing night time diapers, not pull-ups, until when she was 5. And, it was my daughter who told us she was "ready" to just wear underwear at night... she knew herself. And she was right. We were wrong. But even then, there were still occasional accidents. Its okay.
STILL accidents will occur. they are kids. Just use a water-proof bed pad under him. Amazon has many kinds. That is what I use with my kids. I have 4 of them to change out and rotate. Then you don't have to wash the ENTIRE sheets/bedding each time, nor re-do the bed. Just change our the bed pad. I put the bed pad directly UNDER my kids, not in between the sheets.
My daughter is 7 now... and still may have an accident here and there. No biggie. And sometimes she just sleeps so deeply.
If your son is wetting the sheets 3-4 times a night.. then he is not ready nor does he have the biological capacity to be dry all night.
I would, use night-time diapers, instead of pull-ups, because it holds more and leaks less. Huggies make them.
All the best,
Susan