The growth happened really fast and now we saw that it turned brown on the inside while spreading very fast outside. I want to remove the brown part they outer section is the extension of burnt section. Looks ugly if I cut too much.picture to describe the situation
The brown are old vines or vines that are greener further up. I don't know your area of the world, but I do know morning glory. You can go in and cut it all the old brown growth. You unfortunately damage some of the green growth, but it will recover quicker. When you cut into a limb/stem that is brown through out cut it to the ground. If it has some green it, cut it as far back as you want to see new green leaves to grow. No more than a few inches from the ground.
If you are in a warm hot region the growth will have soon. If it's your winter and it's cold out, do not expect it grow back until spring begins.
Your morning glory will thank you for the haircut. It will come back stronger than ever.