"I thank you."
Translation:Jeg takker dig.
As a native speaker I'd have to agree. The only time where I would think this phrase is natural is in writing, like "Du giver mig en gave. Jeg takker dig og nyder bagefter juleaften med familien" ("You give me a present. I thank you and then enjoy Christmas Eve with the family")
I.e. not something you would say to someone's face, but rather read in a book written in present tense.
"Du" can only be used when "you" is the one doing the action. "Dig" is used when "you" is the one the action is being performed to.
Compare to the English "I" and "me" in the first person ("jeg" and "mig" in Danish respectively). "Du" is the second person equivalent of "I" and "dig" is the second person equivalent of "me".