I've seen many browsers, both mobile and desktop, show a warning dialog when I click on an HTTPS link. Just warning of the fact that this is an HTTPS link.
For example, my Nokia E75 asks "Opening a secure connection. Content cannot be seen by anyone else. Continue?"
Why do they do this, as if HTTPS was something to be avoided?
Why? Why? Why?
"Which version of browser is it?"
I've seen many different browsers do this. Any of them.
"What specific link are you following?"
It happens with any HTTPS link.
"Didn't you ask the same question recently?"
I did, but it was open to interpretation. I got some good answers back about some of things that can go wrong with HTTPS that could cause a warning. Rather that bury those answers, I re-wrote the question to fit the answer and I'm re-asking the question I really wanted to ask here.
Best Answer
They want to make you aware that you are using a secure connection.
People have been trained for a while to look for "the lock icon" or something similar in desktop web browsers whenever e.g. entering credit card information when buying something, or visiting their bank's web site.
It appears from your question and comment that (aside from the "tutorial type" message you remember from older versions of IE — and really everything you did for the first time, e.g. submitting a form, was accompanied by a message like that) you notice this behavior primarily in mobile web browsers.
They have less display space to permanently display icons or EV certificate information that indicate a secure connection and therefore need to use other mechanisms, e.g. dialogs, to make you aware of this.