Like Jassy, this is a relatively new issue. I hate to say it, but it seemed to coincide with my Mavericks update. I doubt it's because of that, specifically, but it was around the same time. I really think it's some kind of Safari/iTunes issue.
I actually don't understand what you mean by an active internet connection being the problem. I'm only on line when my Mac is turned on and (I assume) when I'm on email or browsing the web. I don't have any connections set up when I'm not actively on line. But even so, I don't know wh that would cause me to have to log in to iTunes (four times in rapid succession) every time I clear my browser history.
That seems to be the one thing that causes it though - clearing my Safari History. If I don't clear it, iTunes doesn't require me to log back in. Only when I clear it does iTunes make me do those four log ins.
Hope that helps clarify this. I'm not a real whiz at this.