What does the word “s***storm” mean exactly?

Solution 1:

A shitstorm does not necessarily include either a physical altercation, or a dispute. Instead, a shitstorm is any extraordinarily chaotic situation where there is no possible positive outcome.

I don't have a source to back up my claim, but I suspect that "shitstorm" is derived from the idiom "The shit has hit the fan."

Solution 2:

In British English, most likely the equivalent of "a dog's breakfast".