This is an old tree in my parents yard that hasn’t been well cared for. Branches broken under their own weight and left in place, etc- hasn’t grown tall as a result- just thick with much dead wood.
This is quite normal for the tree. All of these symptoms indicate that a very heavy pruning is in order.
These articles will help:
https://www.gardeningknowhow.com/ornamental/trees/mesquite/mesquite-tree-pruning.htm