One could argue that Hamlet murdering Polonius in act 3, scene 4, is a significant turning point in the play for several reasons. During the scene, Polonius spies behind an arras as Hamlet criticizes his mother's incestuous marriage to Claudius. When Hamlet hears something behind the arras, he draws his sword and thrusts it through the tapestry, killing Polonius. Hamlet had initially thought Claudius was spying on him and is surprised to discover that he has killed Polonius. Polonius's death drives the plot of the play and sets several key events in motion. Polonius's death gives Claudius an excuse to send Hamlet to England, where he will attempt to have him decapitated. Claudius has been searching for the perfect opportunity to get rid of Hamlet and has finally found a way following Polonius's death. In addition to Claudius's decision to get rid of Hamlet, Laertes desires to avenge his father's death. Polonius's death also pushes Ophelia to her breaking point and she ends up committing suicide. Hamlet manages to survive his trip to England and returns unscathed. Claudius is then able to manipulate Laertes into fencing Hamlet using a poison-tipped sword to get revenge. If Hamlet had not murdered Polonius, Claudius's plan of sending Hamlet to England would have been interrupted and he would have more difficulty convincing Laertes to spar against the prince.

Well, Hamlet changes throughout the play, so you could argue that there are many turning points, not just one. However, if I had to choose just one, I'd say it is when the king reacts with guilt to the play within a play. Before that time Hamlet had been waffling, trying to figure out what he should do. (Think of the great "To be or not to be" soliloquy.) After that confirmation of Claudius's guilt, however, Hamlet only hesitates to kill Claudius when he is praying—and he does kill Polonius soon thereafter. When he does, he thinks it is Claudius, and he accuses his mother of Claudius's crime, which ups the stakes markedly. After that, Hamlet is sent away—but returns—and things unravel quickly, with a lot of action.
Greg