Although there have been a few episodes with more than one patient (typically the second patient's treatment leading to the "a ha!" moment for the first one), you're correct that most episodes of House focus on a single case.
That is probably a bit unrealistic as a long-term trend. Indeed, there were several points in the show where House's budget was called into question, or cut, though he always managed to get around them. There are some mitigating factors here that help explain why this is so:
First, Princeton Plainsboro is a teaching hospital; the primary purpose of the hospital is to train new doctors. In a couple of episodes we see House teaching medical students (though he doesn't seem to like doing it), and it's probably reasonable to assume his other doctors do as well. It also means that the hospital is more flexible in allowing House to operate his department on the theory that he's "teaching" a team of diagnosticians how to be brilliant.
Second, the hospital also runs a free clinic, and we routinely see House and the other doctors doing clinic work in later episodes. The rest of House's team are also specialists, which makes me suspect that off-camera they do work in their specialty department as well (though I can't think of a specific case where we see this happen on-camera).
But most importantly, the conceit of the show is that House is the most brilliant diagnostician in the country. He only takes cases that are "unsolvable" by anyone else. The key here is that people choose to go to Princeton Plainsboro specifically to see House and his team. The ailments that they are treated for by House are complicated and frequently life-threatening, and require concerted attention by his team to figure them out. That's the kind of thing that the hospital can charge a lot of money for. It also tends to attract rich private donors (we saw more than one of those come through House's care) that are willing to help the hospital out after House saves their lives.
You'll note, in particular, that at one point House leaves the hospital at the end of one season. Even though Foreman and the others on the team are more that capable of running the department, Cuddy wants to close it down because Foreman just doesn't have the name recognition that House did. She flat out tells Foreman that the only reason PPTH has a diagostic is because House is there to run it.
In any "real" hospital, or even in House but with any other doctor, there's no way a team could be as selective as House is and stay operational.