I just went poking around a bit on my Fire 8 . . .which updated on the 16th to 184.108.40.206. Noting appeared to work much differently but when I tapped to open the browser there was a notification about suggestions . . . I'm guessing that's where your issue is?
It looks like, if you have any web page open, you can swipe from the right edge
and have the Instant recommendations pop up. If you swipe in from the left edge you get a different menu -- same as what you get when you tap the three line menu icon.
If you close all web pages/tabs you're at "Silk Home" which has 5 page options. Those you can shift between with right and left swipes.
If you're on 'most visited' you'll see recent web pages you've visited. Below that it says "suggested for you". At first it had a thing about how it would suggest sources and keep them sort of queued up so they'd always be ready, but that went away after I clicked "learn more", saw what that said, and went back. As it is NOW, I see the most recent web pages I've visited -- which I can remove by touching the three dots icon and -- and then a list of 'suggested' articles. To the right is a settings icon.
When I tap the settings icon it shows several sources which I can block. At first it showed 4 . . .then it showed 2. I blocked the two it showed, and went back and still had article suggestions. So I tapped the settings icon again and it showed the other two and I blocked them. NOW . . . 'most visited just shows web pages I've actually been to and nothing down below. If I remove all the pages there's a generic 'go explore the web' thing with links to Amazon, You Tube, Bing, etc.
So that's what I've found. If you've blocked all the suggested sites, the instant recommendations available on a web page still works, but it's based on some other algorithm. I went to kboards main page and it's showing me kindle related searches, no related articles (I guess 'cause I blocked sources?) and Trending News, which is fairly generic and not sorted in anyway. I can scroll that side menu up to see more.
It appears to work the same whether in portrait or landscape. Short swipes in the middle of the page don't activate the side menus -- so if, for example, you're at Amazon, there are sometimes rotating product notices at the top. Short swipes let you flip through without problems. Even a longer swipe works so long as you don't start
at the very edge of the page.
As far as I can tell, it's ONLY an issue if you're in the Silk Browser, and only if you're on a web page. Since most web pages are navigated by scrolling (vertically) or tapping, I'm not anticipating a lot of issues. But, then, I don't use the Fire that way that often.
You could go in and change your privacy settings. You can turn off the 'search and URL suggestions' option -- I think that's the thing that shows you suggestions when you start typing into the search bar. There's also a "Do Not Track" that you can turn on. And, of course, you can clear your history, cookies, and cache. There's also 'private browsing'. But none of that is likely going to stop the 'Instant Recommendations' from being available when you're on a web page, as it's apparently a new feature.
Interestingly, having 'blocked' the suggested sites earlier, I'm now not able to get back to that setting to 'unblock' them. So that's . . . odd. . . . . and there isn't anything obvious in the Help section anyway.