[UPDATE: March 28: A possible explanation for what's going on has surfaced, and suggests the bug is related to the Shared Web Credentials feature that allows apps and websites to share login info.]
A new and annoying bug in Apple's iOS 9.3 update is preventing users from opening links in Safari, Messages and Mail apps, and some third-party apps, as well as causing Safari to crash. Fortunately, there's a temporary fix that will help you restore some of the lost functionality until Apple rolls out an official patch.
ZDNet has tested this fix and can confirm that it offers temporary relief. However, it should be noted that it doesn't restore hyperlink functionality to apps other than Safari (so Mail and Messages and so on are still affected), but it does allow hyperlinks to work again, and also appears to prevents the browser from crashing.
This is a weird bug. Testing suggests that it's not an issue on clean installs on iOS 9.3 (some claim the issue exists on iOS 9.2.1, but we have been unable to confirm this), but instead seems to be triggered by the installation of certain, and as yet unknown, apps. The bug also appears to be sporadic, affecting only certain devices.
Numerous threads have been posted up on Apple's support forum by frustrated iPhone and iPad users complaining about the issue, but as yet there has been no word from Apple either confirming the bug or giving a timeline for a fix.