Turkish Coffee can be a little gritty but properly brewed Turkish coffee shouldn't be overly gritty and if so it was likely not brewed or prepared correctly.
Turkish coffee grinds should be finer than an espresso grind and the coffee should be covered and left for a while to allow the Turkish Coffee grounds to settle to the bottom of the coffee pot.
If you've made Turkish Coffee and it was gritty it's still okay to drink but it shouldn't be too gritty after making it.