Deleting my history fixed it for me.
Safari > Bookmarks icon > History tab > Clear
Something in my history was poisoning the search results, and causing Safari to choke whenever you trying to type something into the search field. Since Safari tries to suggest pages that you’ve visited before, if a bad site shows up there, it can (presumably) cause things to go wrong.
In my case, the problem site was particularly bizarre. I dumped a copy of my Safari History using iExplorer before I wiped it, and started looking for unusual entries. These two stood out to me:
Title URL Visits Last Visited
Cannot Open Page <long Facebook url> -7 14 Aug 2015, 17:16
Cannot Open Page <long Facebook url> 1 14 Aug 2015, 17:12
The Facebook URL is an authentication page that was apparently launched by an iOS game. I don’t understand how it can be last visited in the future (that date is nearly two months after the problem started) or have negative visits.
I’m genuinely mystified as to how that item entered my history – I barely play the game in question, and I’ve never messed about with the date/time settings on my phone. And I have no idea how you get negative visits – possibly some sort of overflow somewhere?
I think it’s likely that this item was the culprit, but how and why are total mysteries.
I guess we should mark this one down as
"El Capitan & Safari 9 fixed it"
I no longer see anything like the 'lag' I was getting before & all behaves as it should.
Best Answer
I had exactly the same problem, and after disabling extensions and restarting Safari it went away. I then looked more closely and found that I had set the letter lowercase-a as the keyboard shortcut to allow flash in ClickToFlash. Apparently this made it not work as a character in any text boxes in Safari! I now have all the extensions back on but without that key shortcut, and everything is working fine!
BTW, this is clearly a bug in ClickToFlash, since the default key to make flash full-screen is lowercase-f, but I can type that fine in any text box.