Large class sizes
One of the reasons parents send their children to private school is for the individual attention small class sizes afford. You want your child to interact with her teachers. You don't want her to end up hiding in the back of a large class room. She's in private school because she wants to learn. So do her classmates.
When you have 12-15 students in a class, you really can teach. Discussions, analysis and explanations are much easier to facilitate with a smaller group. Everybody's opinion matters. From the teacher's perspective it is much easier to assess progress when you are teaching a small group. You can make eye contact with everybody instantly and determine whether they've got it and you can move on or whether you need to go back and reteach.
Small classes play to the reality that everybody learns differently. Students receive information in different ways. They process it differently. The skilled teacher is always assessing where her students are vis-a-vis the material being covered.
The other factor which is so important for effective teaching is that the class is homogenous in the sense that the students want to learn, they are not embarrassed to excel and they...
Perhaps you are considering homeschooling. While homeschooling is doable, the onus is on you to track everything and make sure all the paperwork is completed and submitted and approved by local and state authorities. It's a lot of work. Indeed it is a full-time job. Now contrast that with the kind of life and activities which your child can have at private school: