The term "Adalet Dergisi" means "Justice/Law Magazine/Journal" so it is
not connected to what you are referring. (BTW please notice the correct
spelling.)

If what you mean by "retrieving a case" is "initiate a new lawsuit" or
"learn/get documents/make writings about your case" then what you have
to do is to resort to "Mahkeme Kalemi" (=Mahkeme Yazıişleri Müdürlüğü")
which is a kind of Court office where all official writings of that court has
been done. Every single court has an independent "Mahkeme Kalemi"
which works under the control of that court's judge.

As for abbreviations:
"Mah." is a cummonly used abbreviation for "Mahkeme" (=Court) so is
"Yarg." for "Yargıtay". The other words dont have official abbreviations
but I would like to correct your spellings. Here are the correct spellings:
(In paranthesis included the versions with no Turkish special characters)
"Anayasa Mahkemesi"
"Temyiz Mahkemesi"
"Yargıtay" (Yargitay)
"Danıştay" (Danistay)

There are lots of Turkish Law Periodicals: Yargıtay and Danıştay have
their own periodic magazines, which are published in every 3 months. And
also nearly every single Bar of the cities have their own law magazines:
Istanbul, Ankara and Izmir Bars have the best ones though. Other then
their official Bar magazines, most bars publish other law magazines with
specific law subjects according to their expertise commisions, such as
Criminal Law, Contracts Law and etc. Many law schools all around the
country have their law magazines too.

Hope this helps..