Safari on iOS 9.2.1 / 9.3 won't open links
Update to iOS 9.3.1 or later to fix this specific issue.

Details of iOS 9 updates are posted at https://support.apple.com/kb/DL1842



If you want to try to fix this before Apple patches the bug, here are some steps to try:


https://bencollier.net/2016/03/how-to-fix-ios-9-3s-broken-safari-links/
https://infinitediaries.net/how-to-fix-ios-9-x-safarimail-link-bug/


The latter is a bit longer and more humorous and has some nice photos to look at while you do the prescribed dance. The former link is shorter and more to the point.

Both require the device to be put in AirPlane mode and connected to a computer to resolve the data corruption caused by some apps and iOS not handling the app configurations in a robust manner.



There's a lot of evidence that Safari is in fact having a lot of trouble for some people. I haven't seen a device with issues, but if I did I would do two things to help narrow down the possibilities:


Verify your personal backup situation. Run a manual iCloud backup and make sure it finishes. Run a manual iTunes backup if you do that as well.
Connect the iPhone to iTunes and run a sync - be careful to read and cancel any sync if iTunes warns you that the sync may erase any data or content. The sync is to gather logs from iOS so you can look them over on the computer - but it's not worth losing music files if a sync will erase that.
Keep a log of when you run into issues - you'll want to note the time, the app in question (if you aren't using Mobile Safari to browse links)
Keep track of which apps you uninstall/install while sleuthing is ingoing.


Now - I would recommend anyone contact AppleCare if they can - sooner rather than later as Apple's paid support will a) know how to use these logs and get them to engineering b) know if/when a fix is out c) allocate resources to nail this down based on support costs and volume.

I believe someone has isolated one third party app that is causing Safari on iOS 9.3 to behave badly:


@stroughtonsmith > Reproducible: installing http://Booking.com ’s app will break link handling across iOS. Uninstalling it will fix it
@stroughtonsmith > The fun part, however: it won’t be fixed immediately after uninstalling. There’s a timeout before it’ll fix itself 


If you have that specific app, you might consider deleting it and then waiting a bit (10 to 15 minutes) to see if this resoles your issue. If this is in fact the case, my suspicion is that Apple can work with developers to fix this or release a patch.

If not - you might need to keep track of and troubleshoot the following items:


watch the system log on iOS when Safari is crashing or mis-behaving with cfgutil syslog and Apple Configurator 2's automation tools.
Safari content blockers
Whether some or all web sites are problematic
Safari settings (it's not looking like javascript is failing - but that could conceivably cause an issue like this combined with other settings and internet access)
How you connect to the internet (public WiFi, work WiFi, cellular and which carrier, VPN to a data center or specific other network)
It would take time, but you could easily make a backup and then erase all content and settings to verify that the issue is one or more third party apps. Depending on how full your iOS device is - a restore overnight or over lunch isn't a bad solution to knowing what precisely is the cause.