[jlgaddis]

> ... except that serverB is only listening on a private IP and I want to connect to it in by web browsing (http) ...

Still possible, with ssh's "-D" option that basically turns a remote SSH server into a (HTTP) proxy server!

    $ ssh -D 4444 serverA.example.com

Log in and keep this session active.

In your browser (I use a separate browser, permanently configured this way), configure it to use a (SOCKS5) proxy server for all traffic. The proxy server should be "localhost:4444".

In your address bar, navigate to "http://serverB.example.com". The HTTP traffic is sent over the SSH tunnel to serverA and from there it goes out to the network to its destination. To the destination server (serverB), it appears that the request originated from serverA and that's where the response will go (at which point it is again encrypted and sent over the SSH tunnel back to your local machine). Visit http://icanhazip.com (or a similar site), for example, and you'll see that, to the remote web server, it appears that traffic is coming from serverA.

This isn't the most eloquent explanation, sorry. Look into "SSH socks proxy" and you should find much better explanations. It's really much easier than it sounds! =)