There is a serial killer, psychopath. He kills her(Daphe's) mom.
He stand in court as a suspect. But the court bring in a verdict of not proven.
So Liam proves his superpower in the court and also proves that the psychopath is serial killer. この表現は自然ですか？

There is a serial killer, a psychopath. He kills her(Daphe's) mom.
He stand in court as a suspect but the verdict is not guilty due to lack of evidence.
Liam uses his superpower in the courtroom to prove that the psychopath is serial killer.

I couldn't really change the beginning since the context is not clear . You can say the part about court 2 ways :
The verdict was not guilty
or
The jury(or judge) ruled not guilty
I would also avoid starting a new sentence with 'So'. It is usually in the middle of a sentence or thought as a way of bringing a conclusion to something said before.