Two problems you're most likely encountering.

1. UA doesn't codeshare on certain TK USA-IST flights. Specifically I know LAX-IST. This means that when searching for revenue flights LAX-IST, on UA, many options won't show up. Either book using a different venue (I.e with an OTA or directly with TK) or there's a separate thread on how to book *A flight on UA (one way is pretend to book an award flight and before the final page it will ask if you want to book using $$ instead).

2. Not being able to book an award flight for your 2 year old most likely has to do with the fact that the computer thinks he/she would be an unaccompanied minor. Might have to call in to book.