We don't have kids yet, but we don't pack nearly 2 outfits per day. I probably pack about that many shirts, but I just pack one pair of shorts or pants for each day. Unless you go when it's especially hot and humid, bottoms don't really get dirty like shirts in less than a day.
When we go when the weather is cool, I try to pack as few bulky layers for cool weather as possible. For example, I'll pack 2 sweatshirts for the whole trip, and I'll just wear a different t-shirt underneath the sweatshirt each day if I end up needing it that much. Of course, your mileage may vary with kids, since they're more likely to spill.