I'm a native English speaker as well. I typed the same thing because I was also going for a literal translation. In retrospect, I don't think it's a valid translation. Here's why:
Usually when we say, "I think to…" in English, it has a different meaning that implies, "I got the idea to…" For example, "I didn't think to close the gate before letting the dog outside."
So, I guess you could make a case for, "I think to know it," being a valid English sentence. For example, in this context: "John thought to know his manners well before going to the dinner party."
But the meaning is totally different from the Dutch sentence, which makes it an invalid translation in the end.
It means something different. "I think to know" would mean that you are thinking in order to know something - or that you could never know it without first thinking about it. "I think I know" rather suggests that you are uncertain of your knowledge but it is more likely that not.