SoCal does, yes; about half the water going through the SWP from NorCal, or ~75% if you include Bakersfield/Kern as part of SoCal (though most would consider it Central Valley).

But SoCal isn't only LA. LA itself gets a bit less than half of their water from MWP, which manages the water from the SWP and the Colorado. About the same amount it gets from the the eastern Sierras. These are supposed to drop to ~10% of LA's water supply as recapture/recycling projects complete.

Or computed the other way around, LA only has rights to ~20% of the water managed by MWD. Of course water supply, distribution, and rights are all blended and traded around all the time, but generally speaking it's not "LA" using up that water from NorCal, the consumption is significantly more from the cities and farms that came after.