You can have Turkish breakfast on your own, it’s just the big serpme spread that some places only serve for two. Look for “kahvaltı tabağı” on the menu — usually cheese, olives, tomatoes, cucumber, jam and a few other bits on one plate. If you really want the little dishes all over the table, ask whether they’ll do a smaller spread before sitting down.
But you don’t have to do the whole spread either. Menemen with bread and tea is a perfectly good breakfast. Or börek from a bakery, which is what I’d go for if I wanted something quicker. Around Kadıköy you’ll find plenty of bakeries selling poğaça, açma and simit too. Go somewhere that’s baking and selling throughout the morning; those pastries are much nicer fresh than after sitting around.