What are the reasons for considering the murder of Polonius as the turning point in the play Hamlet?

You are right in identifying that Hamlet's murder of
Polonius is a very significant turning point in the play. Firstly, let us remember that
this is Hamlet's first act of open violence in killing somebody. Following "The
Mousetrap," now Claudius understands the true danger that Hamlet represents to his
person. Note what he comments to Gertrude in Act IV scene
1:

His
liberty is full of threats to all,

To you yourself, to us,
to every one.

Claudius
realises the way that Hamlet's presence threatens his own life and power, and continues
to act to do away with him, and thus the murder of Polonius represents an escalation in
the conflict between these two characters.

In addition,
this scene is important because of the way in which Hamlet confronts his mother after
killing Polonius, and makes her see the impropriety of her actions and doubt her
husband. This represents the first break in the relationship between Claudius and
Gertrude, who up to this point in the play had been united in purpose. Gertrude in Act
III scene 4 agrees not to sleep with Claudius any more and obviously begins to side with
Hamlet.

Lastly, let us remember how the murder of Polonius
impacts Ophelia and Hamlet's relationship with her. The madness of Ophelia as
demonstrated in Act IV scene 5 and the way that her father's death at the hand of her
lover has made her "importunate" and "distracted" is of course key in the way that
Laertes comes to side with Claudius against Hamlet, leading to the final tragic duel
that kills them both.