I am curious how high the water table is. Those bushes are the "salt cedar", growing with their roots in the saline water below?
You are probably aware of Geoff Lawton's "greening the desert" project, seems like someone referred to it above, talking about his use of waste agricultural water. If you by chance are not aware of it, I recommend you look at it. Sorry I can't give you a link.
I agree with the idea of waiting and watching, becoming familiar with your place before you begin your work.
One permaculture adage is that the best place to store water is in the soil...
If you do decide to make swales, I think you would want to fill them with wood (brush) chips. Those plants are green, and they are getting moisture from somewhere... maybe that saline ground water.
What occurs to me is to try to increase carbon. If not already familiar, seek out Elaine Ingham's work. Soil food web, life in the soil. You are in extreme conditions to be sure, those salts in the water and soil can be somewhat mitigated by the soil microbiome community. Get more carbon into the soil.
Maybe I am just making noise here, but if you could even get newspaper and raise worms,they would transform it into more useful compounds.
To get some cleaner water by evaporating (solar still
) the water out of the salts. Don't know yet what you would do with the salts you accumulate, but they are minerals of some kind, possibly some use for them.
I also have an idea that rather than try to rehab the whole plot at once, you might be better off to start small... under just one of those bushes, apply the water you get from the solar still and the worm compost, and mulch over it with wood chips or newspaper of some kind.
Think of it as apilot plot. If you can get some more life, a richer more diverse community in just one spot, it may spread as it creates its own conditions.
If you only apply that fresh water to the one location, and prevent evaporation from that space, then the moisture will flow out from there, carrying some of the salt with it. If you create conditions at the perimeter around your test plot that discourage evaporation then the cleaner water will flow further out. But, I don't think plastic would be a good way to prevent evaporation... something that lets the ground beneath breathe....
so, maybe I am just dreaming, but these are my first thoughts on hearing of your project. good luck, and get ready to be very very patient. It's an ambitious undertaking, no overnight fix for this.