My normal hair on the top of my head is growing back very slow and thin is this normal? I had FUE transplant 1 month ago.

It can take 6 to 12 months after surgery to see results.  If you are concerned it is best to contact your doctor

Your normal hair is likely undergoing shock loss. It should eventually grow back, give it a few months before worrying too much. Generally the drug Finasteride helps immensely with preventing this but if you've already lost your natural hairs all you can do now is wait for them to grow back.

It is expected that you will undergo a shedding phase after after your hair transplant, and you should experience regrowth of your hair beginning in the third month post procedure. At this time in your recovery, your hair will look its thinnest so hang in there and continue using your postoperative medications.

Hair does change a bit after a hair transplant. I tell my patients not to pay much attention to changes in hair unless there is shock loss, something that can be prevented, see below

Hello, the remaining hair are probably at a shock loss stage, so they behave like transplanted hair, they will grow starting 2 months or so
Dr Amit