I too have the problem that links in Safari no longer work after iOS9.3 update. iPhone 6, 128GB
It's so good that people make suggestions like 'disable Javascript', but really this is not a solution. Half of the sites you visit will probably not work properly. And no, the old faithful 'Reboot your phone' also doesn't work, so pls stop suggesting it - or find a job in Windows Support.
Suggestions like 'Use Chrome/Firefox' are also just a distraction - they are using the same underlying browser/webkit. So although (for example) Chrome will allow you to more easily open links in a new tab, this also isn't a solution. Clicking on links after searching simply will not work.
And of course then we get the 'Restore/Reinstall' iOS9.3. Really? This is rather akin to replacing the engine in your car every time you run out of gas. And worse still: It won't solve anything. In various forums this has been confirmed (usually at the cost of spending a couple of hours at a Genius Bar). This appears to a be a new bug, introduced by Apple engineers. It is almost incomprehensible that the engineering team didn't spot it as part of their QA, as similar issues were reported with various beta releases.
I hope that instead of enjoying a great Easter break, they are slaving away on iOS 9.3.1. With fixes. Producing poorly controlled iOS releases (thank heavens I have not yet tried to update any of my 'older' iOS devices) damages the reputation of Apple, and turns fairly expensive digital devices into shiny coasters. If - like me - you use your iPhone / iPad extensively as browser - it needs fixing, and fast.
No: my post doesn't provide a solution. Only one organization can do that: The Apple engineering team. Get to it.