If you are curious to know the roots of the words and more phrases in Turkish to say thank you, check this out !

Türkçe teşekkür etme yolları (how to say 'thank you' in Turkish) :)

The most common way to say thank you in Turkish. 'Teşekkür' is an Arabic word as you can separeta the consonants such : -ş, -k, -r and it also can mean 'şükür' or 'şükran' all those mean thanks more like thanks to god.

As it is the same as teşekkür ederim, it sligtly less formal or shorter, literally translation is 'thanks'
şükretme(arabic): thanking to god

It is a funny and friendly way to say thank you. If you use this phrase to a Turkish speaker they will smile. It is not anything formal, friends use this phrase with each other or foreigners cuz its funny:)

This phrase literally means 'health to your hands' and it is perfectly accurate to use when someone cooks for you. You can say it to a friend or a family member, after they serve the dish or in a restaurant.

This is sort of an old school way to say thanks. Minnet is an arabic word and means when you owe to someone for the favor. You owe them from your heart. 
Çok: very

again an Arabic word means this is on you, this is your nice heart and thought and ideas, evaluation. When someone complements you, you might answer, teveccühünüz. 
Vecih: the face

very simple way to say thank you in Turkish, informal but also looks like you spend less effort to say thank you either you don't have time or it was not a big deal. (sometimes, not always)
sağ olun: more formal

this is the one almost nobody uses from the X generation. Very old saying that was used in early 19th century. 
Hürmet (arabic): Respect 
Hürmet etmek: to respect the opinion or so.
Bilmukabele (arabic): likewise