iPhone won't connect to captive portals

See-->Use captive Wi-Fi networks on your iPhone, iPad, or iPod touch - Apple Support

If you don't have any luck with the above link Lawrence has posted, then I can only suggest you try and remove all references to connection for that site/wifi, make sure the Content Blocker is removed and clear Safari cache by going to Settings>Safari>Clear History and Website Data, then "Reset Network Settings".

You made this remark previously "It is connected to the wifi network in Settings, but Safari doesn't connect to any urls" which can be a common network issue and the usual fix is "Reset Network Settings" from Settings>General>Reset and/or Reboot the Router.

Obviously you can't do the second if it's not yours but the first might be worth a try. I take it this loss of internet while having a wireless connection doesn't happen on your home network or cellular because if it does then the issue might not just be about Captive Portals but more to do with a wider network problem.

If you are having any other network issues I would jump straight in and do a "Restore as New" via iTunes, it's a bit of a pain but considering the time spent so far might be the quickest way in trying to fix this problem. Don't use a BackUp, for the time being anyhow, and just see how a "clean" device performs, if all ok you could then consider returning your previous SetUp by using a BackUp but you also run the risk of reintroducing the problem.

A clean device might not be all that of an issue nowadays because of iCloud and syncing services, once signed in with your Apple ID you may find most of the important stuff syncs back (as long as it's on your other devices to do so) and all you will have left is to download a few apps and setup a few wireless and bluetooth connections.

Could you be running a Content Blocker or any other software that “filters” Safari by any chance.