One more translation:

I promise you know me. 

I think you could say :

Sen beni bilirsin, sana söz veririm.  

But I think there is a better, more advanced way to say it as well...

1. If the person promises that the other person knows him/her:

Beni tanıdığına yemin ederim

2. If the person promises and says that the other person knows him/her, and the person is trying to convince him for something:

(For example: I promise, you know me. I didn´t do that)

Yemin ederim, beni bilirsin.

PS: In my opinion, it seems that the second one is the one you are looking for. But I didn´t take a risk