What are the reasons for considering that the murder of Polonius is the turning point of the play in Act three (scene four)?
Hamlet has been hesitant in carrying out his revenge against Claudius, putting it off until he can know for sure that Claudius is guilty. He accidentally kills Polonius, making Hamlet feel God is using him as a tool of vengeance to punish Polonius not only for his sins, but to also punish Hamlet for his own sins. Killing Polonius leads Hamlet to look at it in terms of retribution, punishment, and vengeance when he says, "Heaven hath pleased it so/To punish me with this, and this with me."
Now that Hamlet has killed Polonius, he will be considered truly mad by everyone. His mother certainly believes he is, and he will lose Ophelia because of his mistake in thinking Claudius was behind the curtain. After this act, Hamlet has nothing else to lose. It marks the loss of everyone who is important to him, his father, mother, and Ophelia. Though Hamlet doesn't achieve his vengeance with the death of Polonius, he knows now that he can committ murder and carry out his vengeance against Claudius.
At this point in the text, I don't think Claudius really felt himself to be in any danger from Hamlet. Hamlet was doing his "acting crazy" bit for a while, and Polonius had all but convinced Gertrude and Claudius that it was the result of Hamlet's love for Ophelia, a connection Polonius had made her sever because he was sure Hamlet could not really love her. Up until now, Hamlet has been erratic and out of sorts, strange even, but he has not shown himself to be violent to his enemy—his uncle and stepfather. At this point, when Hamlet kills Polonius, he makes it clear that his apparent madness has, indeed, driven him further than anyone had been previously aware. From now on, his uncle will be more on his guard, even attempting to send Hamlet to England to be murdered.