I’d like to keep an eye on my kids’ browsing on their phones and tablets, mainly for safety, not to be super strict. What apps do you use that reliably show full web history and are easy to understand?
For most families, Google Family Link (Android) or Apple Screen Time (iPhone/iPad) are the go-to starting points — they’re free, built-in, and show browsing history in Chrome/Safari pretty reliably without needing any sketchy third-party app.
If you want something more robust, Circle or Bark are solid paid options — Bark is especially nice because it alerts you to concerning content rather than dumping every URL on you, so you’re not drowning in data. 
The main thing: whatever you use works best when your router or the device itself is set up correctly, so apps that work at the network/DNS level (like Circle) tend to catch more than ones that only monitor one browser.