Can a router track a website an Iphone goes to?

This is what I mean. Someone goes on an innapropaite website, will the router track the Iphone going to that website

It could, but that isn't the main function of a router.
A router is designed to route data to and from devices, hence the name.
Most routers have a logging feature which logs all trafic, but it is usually deleted or overwritten after a certain amount of time.

If you wanted to monitor web activity, you would need to use other means, like web content filtering or logging.

Very few know how but most routers can create a log. For most routers, this is small and for diagnostic purposes. Commercial routers could send live logs to other places. Not that 99.999% of people ever care or bother

A router can log everything that goes through it, the phone doesn't matter.

If the iPhone is using that wireless network - then yes. All traffic from all devices on that network flow through the router.

If the iPhone uses the cellular network and your phone data plan, then No. The home router does not know about this activity.

The router also maintains history logs, as does your internet provider. As long as a website doesn't fall under an illegal category in your country, it shouldn't matter if they are storing logs or not.

Your router "knows" because it is your router that is "routing" internet packets between your PC and the internet, and maintains logs, but you can clear the router logs. Once again, in some countries it is "illegal" to delete history and logs.

Yes all internet logs can be found on the router.