I'm not sure what exactly you mean by "stop surfing".
Incorrect, please ignore:
For one thing, the browser should not use any more data after it's finished loading a page. If you're looking at an auto-refreshing page, it should only update as long as you're using the browser app -- not when you switch over to, say, your calendar. Sorry.
However, when you do switch over to another app, the browser isn't shut down, just put on pause. There's no harm in this. Look at it this way: work takes power, so shutting an app down takes more work (=battery) than just keeping it paused (not doing anything). Now, if your phone runs low on resources, it will go around and kill all those paused apps, but (unless they misbehave) it's really not necessary to kill them manually -- in fact that will end up costing you
more battery. (There's a very good thread here somewhere about why you don't need a task killer, but I can't find it right now. I can only find
this, which is very brief and abstract but gets some of the idea across.)
Another thing:
All the apps that a task killer app will show are
currently running (if the task killer app is worth its salt). On the other hand, the "recent apps" list you get with a long-press of the Home key lists just that:
recent apps, whether or not they are actually running. So don't take that as an indication of application state.