In Steins;Gate, aren't the time machine crash and screaming mutually exclusive? When the time machine crash occurs, the time machine is damaged, because John Titor didn't calculate the co-ordinates correctly, so she can't travel back in time, ergo, Okabe can't go with her, and thus he can't scream. But in case of screaming, it was Okabe who went back in time, so the time machine landed correctly, ergo, no crash happened.


Despite this, the first episode shows both of them happening same day.

Aren't these two events contradicting each other?
If I get your question correctly, then the answer is NO. Those do not contradict each other because


   The events you described above (first half of episode 1) happens in the beta worldline. In the beta world line the time machine is complete - hence there was no coordinate error and the machine did not crash. The time machine only crashed on the alpha worldline.


I hope that answers your question.