When making Turkish Coffee you do not strain the Turkish Coffee like you would regular coffee.
Turkish Coffee is meant to have the coffee grounds at the bottom of the Turkish Coffee.
However if you prefer to strain the coffee grounds out of the cup of Turkish Coffee you can do so and still have Turkish Coffee.
To properly make Turkish Coffee you need to have the special pot (cezve or ibrik)
However you could make the Turkish Coffee without the special Turkish coffee pot but having the special coffee pot for the Turkish Coffee helps a lot in making a great cup of Turkish Coffee.
You don't have to drink the bottom of Turkish Coffee if you don't want too but I prefer to drink the bottom of the Turkish coffee myself.