How much do the lawns in Phoenix actually contribute to this problem?

Comparatively very little, when stacked against the massive usage of water by agricultural irrigation. Source: https://conormclaughlin.net/2023/03/visualizing-water-consum...

Using an unsustainable amount of water to grow cattle feed and romaine lettuce in one of the most arid places on Earth is a questionable choice. It seems like the choice will be made for us, soon.

Everything contributes but residential use is a much smaller share in total. Arizona water utilization by class [0]:

   6%  Industrial
  22%  Municipal
  72%  Agricultural