The main conflict in Oedipus is in determining who is he. To put it another way, the main conflict in Oedipus is one of self-discovery. A little background information is necessary. The play opens up with a plague. Thebes is suffering greatly. When the people inquire what the cause of the plague is, there is a curse on the land. Oedipus's job as the leader now is the discover this curse and dispel it.
There is also a prophecy that one would marry his mother and kill his father. As the play progresses, it is becoming clearer and clearer that Oedipus is the one who killed his father and married his mother. Moreover, he is, therefore, the curse that needs to be dispelled to take away the plague.
This process of self-discovery is not easy for Oedipus. This is the conflict. From our point of view, we can the same thing. Sometimes self-discovery is not easy.
it is how the main idea realised in the text