I've googled and there doesn't seem to be any studies to prove or disprove either of our points.
I don't think anyone is saying abuse doesn't happen to children in public schools. I think we are just saying it's less likely to be reported if they are never in public and the physical abuse can be much worse chaining/starving/beating/burning etc which I think we can agree on.
I think that there are two big points. One is that schools tend to serve as something of a safety net. Not only do they put kids in contact with a number of adults beyond their household, but the have also been for decades a convenient way of disseminating and ensuring some basic public health information and practices. Kids have to show vaccination records (at least until recently) when they enrol. If the play sports they have to have a physical. School nurses (less true in recent years of tight budgets) perform vision screens. Kids who are not educated in schools, lose some of that protective umbrella.
The second big point is that public schools are regulated by their respective states (private schools somewhat less so). There are minimum requirements for who can teach, background screening for every adult who is around kids, minimums for hours of instruction, maximum student to teacher ratios and a whole slew of teaching/learning expectations. While corporal punishment is far from being universally banned, in most places there are at least some parameters (witnesses, reporting to parents, prior permission from parents) aimed at protecting children. And there are also a lot of basic health and safety regs (remember fire drills?). Kitchens have be inspected and kept to code, bathrooms have to be sanitary, unsafe conditions (roof leaks, broken windows, etc) have to be addressed.
Now, I will grant that even with all this regulating there's a lotta stuff slips through cracks and crannies. But with home schooling there is very little regulation (all depending on the state) regarding who can "teach" at home (in some states, education requirements are lower for parents than other home schoolers), how much is to be taught and what is to be taught. And while some excel in this free environment, it also opens the door to a range of folks from those who think that learning the Bible is enough to some "free rangers" who believe in just sort of letting kids wander around in hopes that they might learn something. Also folks who think that teaching is much easier than it really is. And then there are the Turpins (and others like them).
California seems to require only two things of home schoolers. One is to "register" as a school with the state. The other is an annual fire inspection. And apparently no one (local school board? the state?) has a responsibility to see that the annual inspection takes place. Can we imagine what a difference it might have made if at least one time a year the Turpins had to clean up the mess and let some firemen walk through to inspect? What if the Turpins had produce some evidence of curriculum planning? What if they had to show books and some student work? What if someone in one of those two states (Texas or California) was empowered to say to the Turpins, years ago, "sorry, the education you are providing for your children falls way below all established standards, do better or send them to school."
This case reminds me of the Stanley family in Arkansas. They were an off-the-grid religiously motivated, patriarchal family of homeschoolers, with a father who defended corporal punishment, and had some very fringy health practices. Enough suspicion that CPS removed all of their children, although they provided intensive services and ultimately returned the kids. Nowhere near what the Turpins are accused of, but still, as I said, enough suspicion for CPS to act. But, we know about them because a bunch of supporters started an online campaign making all kinds of accusations against the local LE and CPS, claimed that the family was being discriminated against for being religious and using "nutritional supplements." Family claimed that physical exams performed on the kids (who had never been to see a doctor) were traumatic. Had a whole lot to say about what they believed kids were being taught in school. And on and on. I bring them up because so many folks were willing to jump on their particular bandwagon and cry about "the state" intruding into families and carrying out some kind of vendetta against religion and home school. Now, their kids might not have suffered serious or lasting harm--or practices might well have gone south with them over time without the intervention they received. But, the folks who said the Stanleys shouldn't have to adhere to any minimums (in Arkansas families are required to register AND bring their kids for state testing once a year), don't seem to realize that they are also protecting parents like the Turpins.
