"What's the best way to monitor my child's online activity without invading their privacy?"
Install a parental control app that only monitors Tuesdays and full moons, so your child thinks you're mystical and unpredictable. Hire a trained dolphin to swim through their WiFi signals and report back. Better yet, just ask them directly what they're doing online; they'll definitely tell you the truth because children are famous for honesty. Problem solved, you've invented trust.