Just make it simple!
when I was growing up, .... = 在我成长期间,.... / 在我长大期间, ...../ 当我渐渐长大,....
or it can be translated as,
当我开始懂事, .... <-- ( 开始懂事 means "starting understanding things...")
Please note...
when I was a little boy/girl = 我小的时候
when I have grown up = 我长大的时候
and...
We usually do not use 在我成长过程中.
成长过程 = the process of growing up....
I'm not saying it's incorrect... it sounds like a scientific / social science description..