Your requirements are conflicting. Think about it. It costs money to build showers and toilets and to provide running water. It's quite unlikely that someone or some organization is going to build a campground with the amenities you want, then let everyone camp there for free, especially close to a major city where the demand will be high.
Basically, you can probably camp for free, or you can camp with amenities, but not both.
In the western US, the rules for camping in National Forests are often quite relaxed. Many Forests allow "dispersed" camping, which is free. That means you pull off to the side of a dirt road, and pitch your tent. There are no toilets or anything else, but it's free. This tends to be in places that don't get overrun as a result, so "not too far" from Los Angeles is less likely. Still, such camping is allowed in other places close to major cities, like in the Tonto NF less than one hour drive from downtown Phoenix. Find the National Forests in the area, and ask them specifically about camping restrictions.
National Forests are also good places to camp in special-built campgrounds for relatively low cost compared to other designated-site campgrounds. State parks tend to cost more. However, this is just a rough guide. The general rules are just that, and they can be quite different near cities.