I have a group of about 20 people and we'd like to get a discount for flying together from LAX to Beijing, and from Shanghai to LAX in March.

So far, we're finding much cheaper prices by booking separately. Am I wrong to think we should be able to get a group discount?

When you look at prices for booking separately, you are seeing small amounts of seats released at discount prices by the airline. Airlines rarely release very many sale tickets so by putting pricing for 20 people, you are effectively making it impossible to access discounted pricing, which is why its you standard pricing (= more expensive).
Your best bet to see better pricing by contacting whichever airline you know flies your route's group department directly. Group price quotes DO NOT show up in online booking engines, even if you put in 20 people. They can only be generated internally at the airline.

Hop this helps!