Translation:When you were a child, you used to read books every morning.
"While" can be used when you have to continuing events happening concurrently in the past. "When" is used to talk about a single completed even that occurs over a longer span of time.
That is what I found regarding this, but i wish I had a better answer. "While I was a child" just sounds really strange and incorrect to my native ears :)
I think that we can't use "while" and "used to" together because, as you said "while" suggests continuation, but "used to" definitely doesn't have a feeling of continuation, although it might have been a continuing event before it stopped.
I'm happy that I finally agree with you about something! "while" and "used to" definitely don't go together.
"As a child you used to read books every morning"
is not accepted (or several other natural variants with "as a child" in them). Reported
"You read books every morning when you were a child"
(which I also like) is accepted.
Also the indefiniteness of the Turkish is really hard to preserve here; it sounds like more than one book per day, in English, but I don't have a way around that.