Proctor under pressure from Abigail's charade about seeing a bird flying in the court, that she claims is Mary's spirit.  Mary accuses Proctor of trying to force her to follow the devil.

Proctor is arrested and thrown in jail along with Giles Corey, who refuses to name the individual who told him about Thomas Putnam instructing his daughter Ruth to accuse George Jacobs of witchcraft.

Reverend Hale quits the court and leaves Salem at the end of Act III, this is a very significant turning point.  Hale becomes convinced that the court is being used as an instrument of vengeance by members of the town and that the truth has been subverted in favor of maintaining the court's authority.

The actions of the characters in Act III, lead up to the resolution that occurs in Act IV.

This is a thought-provoking question, especially in its reference to Shakespeare. A Shakespearean tragedy reaches its climax, or turning point, in Act III, is followed by falling action in Act IV, and concludes with resolution and denouement in Act V. This plot structure was identified and examined by Gustav Freytag, a German novelist, in the 1800s. Since Shakespeare's dramas are structured in five acts and The Crucible is a drama in four acts, some structural inconsistencies must exist. The climax of a drama is the high point of the drama, the moment of greatest interest and dramatic tension. At this moment, the outcome of the major conflict of the work is determined. So what is the final dramatic climax of The Crucible? 

Act III may be interpreted as the climax of John Proctor’s conflict with the powers of church and state that are driving the witch trials; he fails in his attempts to stop the madness. The climax of the play’s most compelling conflict, however, occurs in Act V when Proctor struggles with himself: to die with integrity or to live with shame.

When he tears up his confession, the play reaches its ultimate dramatic climax. At the end of Act III, John Proctor’s plight is far more difficult, but his greatest test of character still lies before him. A final turning point remains.

The tension in the play builds and builds, leading up to the final confrontation between John Proctor and his judges, most notably, the Deputy Governor Danforth. Proctor has made his confession, a lie, stating that he was in league with the devil, so that he might save his own life and go home to his family once again. However, when Danforth demands Proctor's signature on the written confession, Proctor begins to doubt his decision to lie. Finally, 

His breast heaving, his eyes staring, Proctor tears the paper and crumples it, and he is weeping in fury, but erect.

It is in this moment that the play's most significant conflict, Proctor's inner conflict concerning his own goodness, begins to resolve. Reverend Hale yells at him, "Man, you will hang!  You cannot!" to which Proctor replies,

I can. And there's your first marvel, that I can. You have made your magic now, for now I do think I see some shred of goodness in John Proctor. Not enough to weave a banner with, but white enough to keep it from such dogs.

Proctor now realizes his own goodness, that he can be redeemed. He may have sinned before now, but that sin hasn't spoiled his soul, as he once thought. In this moment, when he tears the confession and recognizes his own integrity, we see the play's climax.

Think of the climax of a work as the "Point of no return."  Once the climax occurs, the audience knows how the work will end.  At first, for the audience, it looks like we're going to get our happy ending; Elizabeth has convinced John to lie and admit that he has been guilty of witchcraft.  We know this lie will save his life, even though we know he feels guilty about lying and ruining his name. Eventually, John can't take it- he knows his name is more important than living a lie.

Here we have our climax.  Once he rips up his confession, we know how this will end.  He's guilty in the eyes of the court, and therefore he will hang.

Another major turning point in the play that seems to have been missed in the initial answer is not only Tituba naming names, but even before this when Abigail turns Tituba into the first scapegoat. It is at this moment that we see the weak will be preyed upon and it will also be accepted my the majority who are looking for someone to he responsible for the indiscretions of others. 

Miller we makes this point to show how the HUAC did the very same thing when they needed to blame people for something. They needed to look proactive and or sent in the problem rather than looking as if it was being ignored. Miller's play mocks the moves of McCarthy. Hence why Abigail, who is a representation of the HUAC, blames the slave who she knew couldn't save herself. In addition, Miller points this out in his description of the actions of Abigail regarding Tituba in Act I.

There are several instances where turning points can be seen.  One of the most signifiant turning points would be in the ending of the First Act.  It is at this point in the drama where Tituba panics and "names names."  Once the girls see Tituba receiving public praise for naming names, they begin to do so.  This moment is a significant turning point because the practice of "naming names" and thus adding to the hysteria of Salem becomes evident.

Another turning point would be in the ending of Act III.  The courtroom scene where the girls exert their power over Mary Warren is significant for a couple of reasons.  It shows the power that Abigail and her crew hold over everyone in Salem.  They are able to manipulate the courts, the civil and religious leadership, and essentially work the system to their own benefit.  This was seen in how they get Mary Warren to capitulate and recant her accusation against them.  Another reason it is a turning point because John Proctor realizes that he cannot win.  He understands that the force of goodness present in people like himself, Goody Nurse, and Giles Corey will not be able to overcome the sinister forces of the girls.  It is a turning point because Proctor admits to his affair and confesses that he is not able to be silent and remain isolationist any longer.  This is a turning point because Proctor recognizes the strength of the forces against him and the forces that are running Salem into madness.

The final set of turning point can be seen in the drama's end.  When Proctor initially signs the confession and then recants because of his "name," it is the critical turning point in the thematic development of the drama.  Proctor's goodness, as Elizabeth puts it, and his sense of honor in a world filled with dishonor is a significant turning point in the drama.  It is a turning point because it moves Proctor from an ordinary human being into a supremely transcendent figure.  This is a turning point because it marks the point in which Proctor, himself, changes.  In doing so, he provides an inspiration to those around him and the audience, as well.