Bermuda and Zoysia are two of the few lawns that prefer a shorter mow. The shorter you mow, the less leaf surface is left to manufacture food for the plant. The shorter you mow, the hotter the roots of the lawn. The shorter you mow, the more sun the weed seeds recieve. All three make it harder for the grass to compete against the weeds.
Few people fertilize their lawn enough.
If, when you don't get the growth you need, often the answer is 3 applications of a good low-nitrogen lawn fertilizer. Scenic Nursery recommends Master Pro Fertilizer at 7-14 day intervals. This builds up the residual nutrient level so the grass can really grow and build the rhizomes and stolons that thicken your lawn and prevent weed invasion. After this program, go back to a schedule of fertilizing 4-6 times a year.