Same as most have reported. Ran the cursed 9.3 update Thursday night, just noticed Safari links weren't working last night. All day today (Sunday, my usual day to sit around browsing the web), I tap links, and nothing happens. Hold a link, Safari freezes, or freezes then crashes.
Uninstalled Booking.com (which I heard was a random thing to try) — no luck. Disabled pop up blocker and JavaScript; only succeeded in making links accessible from a Google search in Safari; links in the actual webpages accessed don't work.
Downloaded Chrome: same thing happens, but at least in Chrome when I hold down a link, it gives the option of opening in a new tab. Sorting through the myriad tabs this creates is rather painful and kludgy, though. Once on your desired webpage, some links work, some don't.
I just checked, and my favorite 3rd-string iPhone browser, Puffin, has come to the rescue. Works like a charm. An imperfect solution, but at least I can use the web like a normal human being whilst Apple gets their stuff into gear.
I have an iPhone 6 recently (regrettably) upgraded to iOS 9.3. While I'm glad to see I'm not alone, this is quite a shockingly major bug to have let slip. Apple, shame on you. ::sad face::