When I leave next week I will not come back until next year.

My try

Haftaya bırakınca, gelecek seneye kadar geri gelmeyeceğim.

does to leave mean to give up?

It means to go. (leave a place)

Haftaya ayrildigim zaman bir dahaki seneye kadar geri gelmeyecegim 

Exactly! So I am curious why you translated it as "when I gave it up" instead of "ayrılmak"

bırakmak= to give up

ayrılmak= to leave

 I looked up ´leave´ in the online dictionary and it gives ´birakmak´ to leave something or somewhere.I have seen ayrılmak used for ´to separate´

 Can ´bir dahaki´ mean ´next´ ?

I got it. That´s not your fault but that is wrong. It can be the last meaning that means "leave someone" to finish a relationship only, and it is used with -den ayrılmak: birinden ayrılmak

thank you everybody. I wanted to make sure that I had the correct meaning and you have all been very helpful. Thanks again!!!

 Ok-thank you for correcting me. I´ll remember for next time.

Could I have said ´gitince´ instead?

gitince=gidince

it means when you are gone

lol

 Sorry-I thought it could mean ´when I go´   

It can. It doesn´t have a tense, and personal pronoun suffix either. It may mean anything;

when ...[someone]... go/goes/went/have gone/had gone

Thank you.