Can I say "glad to be treated normally" to express my gratitude?

Someone writes back to me, consoling me that everyone has low-emotion periods and there's nothing wrong about that. I feel quite relieved since she doesn't show excessive worry or view me as someone weird. But shall I express this feeling by using "Glad to be treated normally"?

I'm going to depart from the other diagnosticians and suggest that you "overthink" this a bit. Or perhaps just think it over.

There's a slight difference between normal as applied to you and as applied to others' treatment of you.

Perhaps you mean

Glad to be treated as normal.

Which means that you're happy that others are not treating you as abnormal, i.e., as sick or (as you say) weird.

But

treated normally

can also mean treated in the usual or appropriate manner, and that can be applicable to whatever condition you're in. If you're suffering from mild, situational depression that afflicts everyone from time to time, then "normally" would be slight concern. If you're suffering from involutional melancholia, then "normally" would be checking you into a hospital.

If they were good to you and treated you like you're a normal person, instead of judging you or upsetting you, then it's a valid response.

"Glad to be treated normally"

I'm assuming that normally you are treated in a way that you appreciate.

A better phrase would be (as @deadrat answered)

"Glad to be treated as normal"