Brutus's speech in Act III and his misguided decision to allow Mark Anthony to speak to the mob constitute the play's turning point. At this stage in the play, things could go either way; it isn't obvious which side will prevail. Caesar has been brutally murdered; the common people, the plebs, are angry, confused and fearful. They loved Caesar and so it's essential for the conspirators to quell the masses, to prevent widespread disorder from breaking out. Brutus's speech appears to do precisely that. He goes out of his way to convince his restive audience that he loved Caesar as much as they did; he only participated in his assassination because he genuinely believed that his old friend wanted to make himself king and the people of Rome slaves.

On the face of it, Brutus's speech is a success, but the Roman mob is notoriously fickle. Brutus seems not to realize that the people can be just as easily swayed by an opponent. He's allowed the instant acclamation of the mob to cloud his judgement, ironic indeed given his general contempt towards the common people. Brutus has also made the fatal mistake of accepting at face value Mark Anthony's apparent acquiescence in Caesar's assassination. Underestimating both Mark Anthony and the Roman people is a fatal mistake by Brutus, one that represents a significant turning point in the play. Up until now, the conspirators have had it all their own way, but once Mark Anthony finishes one of the most famous speeches in all English-speaking drama, the tables have turned dramatically and there's to be no going back.

Oooh!  Great question.  While I think careful readers of this play could adequately argue for either one of these choices, I believe the greatest point of tension is not Caesar's death (which happens relatively early on and the play is not the play without the consequences of Brutus' decision to allow Antony to live and to speak at the funeral), but Brutus' veto of Cassius' concerns regarding Antony's threat to the conspirators.  Brutus is not the judge of character that Caesar is.  He sees all people for the good in them...with the exception of Caesar's greed and ambition, the reason Brutus chose to join the conspirators.

Antony follows Brutus' boundaries for speaking at the funeral to a "T".  He speaks well of Caesar, and does not speak "ill" of the conspirators, although he repeats, with a different tone of voice, how "honorable" Brutus and the others are.  This repetition drives home that they are NOT honorable, but traitors to the state and butchers to Caesar.  All events unfold accordingly.