Answer: A lilac can be renewal pruned either all at once (cut it off drastically short at ground level all at once, do this in late winter) or over several years' time. I would suggest removing a third of the oldest stems by cutting them off near the ground. Also remove any new shoots that are smaller than a pencil in diameter. Do that now. Next year in late winter or early spring, remove half of the remaining oldest branches. The year after that, take out the rest of the oldest branches. Always cut these at the base. In this way you will renew the entire plant in three years. It is less of shock to the plant to spread it out. It also allows you to enjoy flowers each year.
After that, each spring right after it blooms you can thin a few older branches and remove any new shoots that are smaller than a pencil. This maintains it with an ongoing process of growth and renewal so it never becomes overgrown.
But I am not sure this would explain the browning and poor growth you are seeing. You might want to consult with your professional nurseryman to see if they can diagnose that for you more specifically. Do that before you cut it back. I hope it is nothing serious.
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