How do I save a lilac bush?

It might indeed be your climate. I too am in an arid climate and the true tree lilacs just don't do well. The leaves are too thin or something for our high evaporation rate. I'm assuming you have the true tree lilac and not the common shrub lilac......there is a difference. We can grow shrub lilacs here....better in situations where there is evaporation around them......lawns and not gravel mulch They just don't like the hot air coming off rock or the hot blowing wind........in an enclosed yard where the winds are moderated we do better.

What about the soil? If you are fairly alkaline, that might be a problem. Increasing organic matter, using a not reflective mulch and keeping the area more humid might help. But you are still fighting the pH.

Finally there are lilac critters. There's a borer that gets into the stems, lilac ash borer. Finally it might be a case of tangled roots underground, the plant is strangling itself.