So if you're looking for a solution to "stream" your iPhone screen directly to the Internet, the answer is NO.
But if you are willing to break the problem down to two, that might be possible with an external computer, PC or Mac, though PC is recommended.
Two break it into two parts, you first want your iPhone screen to appear on the Mac or PC's screen. We'll use something called "Airplay Mirroring", and on Mac and PC, "Air Server", or an app on Mac "Reflector". You can find both Air Server and Reflector on the Internet, and both offers trial.
Then you want to stream your Mac or PC's screen to the Internet. On Mac, I DON'T prefer any application, they are all slow and have poor quality. This is what I like, "321ShowIt", which is Java-based, but works better than the solutions here.
There's also a down side to 321ShowIt, it streams to a webpage that needs Java to watch. But those other solutions streams to Twitch, where people need only Flash, and have dedicated apps on both iOS and Android.
On PC, you of course can use 321ShowIt, but I prefer XSplit Broadcaster, it's an amazing high quality streaming application for a wide range of platform, including Justin.tv and so on.
So, if those don't suit your need, a jailbreak will be required. But even so, you will be limited to providing VNC connection. Streaming from iOS device is really not a good option since compressing the H.264 stream and serving it live will cause a noticeable performance hit on even the fastest Apple A6.
Anyway, good luck!
Best Answer
Not exactly an answer, but reading documentation: Share screens using Messages on Mac I noticed:
And:
It should be possible to restrict your child (or yourself) with communication limits (Set up communication limits in Screen Time on Mac) and this way disallow screen sharing. Whether cost of doing it is too high is another story.