It's not actually a matter of "natural"-ness. In this case - assuming you're not stating there's a change in this number - the correct answer is "have".
I have watched 6 episodes of Kuzu no Honkai.
This is current information, so it's present tense.
"Had" in this sentence implies that it is no longer the case so it would be used in a situation like:
I had watched 6 episodes of Kuzu no Nonkai until yesterday... but then I had a binge-watching session and completed the series.
So, now you've watched all of the episodes, which is why you use the past tense.
Similarly, if you had a friend over to watch the show with you, you might have a conversation like:
Friend: Before today, how many episodes of Kuzu no Nonkai had you watched?
You: I had watched 6 episodes.
Again, you use the past tense because this information is no longer current.