To understand what's really happening, you need to know that whenever you try to connect to any site that doesn't support Secure-Socket Layer (SSL)/Hypertext Transfer Protocol Secure (HTTPS) connections, you're going to get an HTTPS error message. Below is an example of a typical error message.
Of course, that's not the error message you're seeing from Bing. That's because Bing is using the Akamai CDN. A CDN is designed to speed up traffic to popular web sites by mirroring these sites on their global network of servers. Thus, when you first "connect" to Bing or other Akamai-supported sites, such as the White House, chances are you're almost certainly connecting to a near-by — in terms of network traffic — Akamai server instead.
Akamai supports HTTPS connections, but only if the host site supports it. Otherwise, the link along the security chain from the Akamai network to the core website is broken, and the connection wouldn't truly be secure. In this situation, Akamai presents a different error message — the one you'll see now if you try to reach Bing or The White House using an unsupported HTTPS connection.
This is also not the first — nor I'm sure will it be the last — time that someone reports a site is broken because of this problem. All that's really happening is that a normally invisible part of the internet infrastructure, the CDN, is appearing because someone is trying to force a secure connection to a site that doesn't support it.
Since FireSheep showed up in 2010, more and more sites — including Twitter, Google, and Facebook — have all enabled secure connections. It's well past time Microsoft started supporting basic security on Bing as well.