Learn Turkish: Seni bile inandırdığına göre, bu saçma hikâye şüphesiz doğru. - Since he even convinced you, this nonsense story is undoubtedly true.

This uses the deductive inference structure from this lesson. It breaks down into the verb root inandır- (convince) + the participle -dık (which softens to -dığ) + -ı (3rd person singular possessive "his/her") + the buffer -n- + the dative suffix -a. Combined with göre, it literally translates to "according to his having convinced," which flows naturally in English as "Since he convinced..."

Why is the verb inandırmak instead of just inanmak?

inanmak means "to believe." Adding the causative suffix -dır turns it into inandırmak, which literally means "to cause to believe." In English, we simply call this "to convince."

I learned that believing takes the dative case. Why does seni take the accusative case here?

You are completely right that inanmak takes the dative case (for example, sana inanıyorum - I believe you). However, when you make the verb causative (inandırmak), the person being convinced becomes the direct object of the new verb. Direct objects require the accusative case, so it becomes seni inandırdı (he convinced you).

Does the position of bile (even) matter in this sentence?