I assume that you are flying in Y.

If you have TK *G, I think you can bring your family in - but I'm not sure because I did guest someone at IST lounge when a guy travelling with his family + his adult brother (he is Turkish so as long as he said someone will guest him in then it's ok). I have seen they turn down an arrogant Swedish woman attempted to enter the lounge with 2 kids - she was more demanding the staff to let them in than asking them nicely - she has SK G - which allows her to bring her family in in SK lounge but not other *G lounge.

I was there and I could guest her child in - but I chose not to because of her attitude.

About Fast track, it's YMMV. You might not even have to go through security at transit.