Oh, yes, that's normal for kids who have just started going to school/preschool etc.
Quite simply, there are hundreds, if not thousands of cold and flu viruses circulating in society. As adults, we've already built resistance to the many we've encountered over our lifetimes, so we won't be carriers for those germs and pass them on to our kids, because the germs can't breed and build up in us (after we've developed defenses against those specific strains).
At school, your child is being exposed to a much, much broader spectrum of pathogens because the other kids they meet also haven't built up resistance, and will acquire and pass along what they run into, and vice versa, so there are huge multiples of kids with very little resistance to anything (and, by definition, the capacity to carry and pass on many more varieties).
The different resistance of other adults they encounter in their households and families will mean other, different germs that maybe your family has not been exposed to before, (which will mean you will also get sick a bit more often), or ones you have been exposed to, but the other families haven't (so, while you won't get sick and pass those along, another family might pass along those germs to your kids that they would not get from you).
Schools and other kids are germ factories. Keep in mind, that, as you rightfully view other people's kids as disease-infested vermin that they are, technically, your kid is the same to them.