Is it a dog walking area?

Turf grass would be a lot easier to manage in that spot. Either way if there is any dog access very short plantings like these are going to be used as a toilet by them.

Dogs are walked by that area, but the patterns just do not suggest dog urine to me. Yes grass would be easier to manage. I had a thick St. Augustine lawn there, but lost it due to watering restrictions. I planted creeping thyme because it is one of the few ground covers permitted by the city of Los Angeles for the parkway, and it it suppose to be drought tolerant.

Can you post a close up of the brown patches so we can see the damage to the leaves?

In my experience thyme gets woody brown areas when it’s time to be divided - advice I got here for wooly thyme that I’ve applied to that and lemon thyme..

Here's the close ups. It seems to happen during hot spells. I'm wondering if it needs more water. I looks like it's baking to me but I thought it liked it hot and relatively dry.

When you plant creeping thyme basically you are rock gardening, with associated chores and issues such as this browning you are now having - your thyme planting is never going to behave like a strip of turf grass. Maybe down there it is just too hot and dry for too long every year for any kind of lawn grass to be left without summer watering but here in "Upper California" (July rainfall total a fraction of an inch) brown lawns are the norm. With them all coming back each fall, once it rains enough.

Hi, I'm in the process of replacing my front lawn to creeping thyme. I read an article online that the creeping thyme will die after 5, 6 years. Is that true? It means I have to replace every 5 years? thanks

Well, I've had mine for over 15. But it does, as I said, die in spots.