from the screenshot it looks like you are not on a search engine but, trying to visit a website which might not provides a secure connection. you can tap on advanced and visit the site by accepting to you are willing to visit an unsafe site, where you might not wanna provide any sensitive info. I hope it answers your question!

The thing is, it does this with whatever site I try to open...

can you provide some of the urls to the site you are facing this issue

I cant even lunch a search