How do I restrict a SINGLE website without turning off private browsing and restricting all adult content?

Not sure if you intended it to sound that way, but it sound like you might be confused about blocking content and private browsing. They are in fact two separate things.

iOS of course provide for setting restrictions, but this essentially turns off access to the browser. Private browsing prevents sites from tracking you by disallowing cookies and not logging browser history among other things.

What you are talking is content filtering. You can try using a content-filtering Webproxy like OpenDNS on your WiFi network, and turning on Private browsing, but a savvy child might still figure out how to remove the proxy or just use the 3G/LTE connection.

You set up a proxy on iOS by doing the following:

1. Go to Settings > Network.
2. Tap the blue arrow to the right of the network that you want to use.
3. Tap the BootP tab.
4. Look for the HTTP Proxy option at the bottom of the screen (you may need to scroll)
5. Fill in the details of your proxy service.

There is no way to do this with the built in browser. There are 3rd party browsers in the App Store; it's possible that one of these has the feature you need. You can install it, then disable Safari in Restrictions.