What do you have now in the green places? Is it bermuda?

For Phoenix you might have another week before it's starting to get late to seed. But if you already have bermuda, why isn't it full and dense already.

How often do you water and for how long?
How high/low do you mow?
When were the last two times you fertilized and what did you use?

Yes, we have bermuda now in green places.

I have been watering twice a week, 30mins a time, in summer. In winter I did not water at all.

Have not fertilized it in a year.

The lawn had a lot of weeds, and had gone through repeated applications of lawn weed killer.

I am posting a picture of the lawn -

Wow, that looks awful dry. I would go to every other day at 20-30 mins. Also need to fertilize every month as Bermuda is a nitrogen hog

Something is wrong with your soil. Watering for 20 minutes, 2x per week, should be giving you continuous soggy soil. Clearly you have a lot of stones going on. Stones heat up in the sun and hold heat overnight causing the soil to dry out more than normal. I'm guessing you also have a runoff problem such that after a few minutes, all the rest of the water goes into the gutter. If you have runoff, then you have to stop watering as soon as you see water running off. Stop for 15 to 30 minutes to allow what moisture is there to soak in. Then start watering again. Continue alternating watering until you get a full inch of water as measured by filling some cat food or tuna cans places out in the yard.