I think your question needs to be addressed like this: there is a difference between what might be called 'normal' in terms of anyone's personal psychological development, and what is deemed 'normal=socially-acceptable' in terms of one's relationship to the society you happening to be living in.
As regards the latter, 'normal' is largely a matter of the agreed-upon values of whatever place you're in: for example, in certain Islamic cultures it's 'normal' for women to wear the burkha, whereas in western society it's 'normal' for women to wear make-up, miniskirts and heels.
In some parts of the Islamic world female circumcision is considered 'normal'; in certain Oriental cultures it was considered 'normal' to tightly-bind the feet of young girls to inhibit growth. In what's loosely referred-to as 'Ancient Greece' it was considered 'normal' for older men to have intimate sexual relations with teenage boys...I could go on.
'Normal' in the societal sense appears to be a flexible proposition; it was only in 1967...1968...can't remember...that homosexuality was de-criminalised here in the UK in accordance with the recommendations of the 'Wolfenden Report'; more to do with expediency than altruism I think, but a step forward nonetheless.
As regards the former point, without wishing to go all 'Freudian' on you I would say that desires such as needing to cross-dress are something that emerge according to one's personal earliest experiences at an 'unconscious' or 'pre-conscious' level; this is to say (with acknowledged brevity) that the desire to cross-dress, for whatever exact reason is something that is a part of you; it's not your 'fault'...something you should feel guilty about.
I get the feeling, Andrea, that you're in conflict between what you feel is 'normal' and what your world tells you is 'normal'.
You're not the first, won't be the last; hope my humble comments are of use.
Jacqui.