How can I supervise/monitor the iPhone of my kid?
Solution 1:
Parental controls are more so about restriction rather than monitoring. Although your intentions seem honest and reasonable, someone could maliciously use a feature like this to spy on friends or significant others.
Explore Apple's robust iOS Parental Controls instead.
Parental Controls
Solution 2:
You could use a 3rd party restrictions service like Qustodio or curbi. These are profile based and allow you some monitoring.