1500 Dollars is around 6000 Turkish liras.
If we talk about Istanbul, you will need to pay at least 2000-2500 TL for rent if you want to live in decent apartment in good neighborhood. Of course you can find apartments around the city starting from 1000-1500 TL.
And you have 2 children that means you will likely to spend 1000-1500 TL to basic expenses like food or clothing.
And for the bills on total 500 TL in winter - 250 TL in summer.
Public transportation for 1 person for 1 month should be around 200 TL.
Having a dinner in a decent restaurant for 2 people can differ around 60 to 200 TL.
Kindergarten ? School ? That might cost you around 2000 TL for 1 child if you use private schools.
So yes you can survive but you will not have a comfortable life with that amount. Not in Istanbul.
Most expensive cities in Turkey goes like Istanbul > Izmir = Ankara > Other cities, I would choose the other cities option if I had the chance.