This is a somewhat difficult question to answer because the path that led to the American Revolution does seem to resemble a gradual escalation of tensions over time. The relationship between Britain and its colonies deteriorated across the 1760s and 1770s, culminating with the Revolution itself. From that perspective, if by turning point we mean the point in which this trajectory originates, I think the answer is quite clear: it can be found in the French and Indian War. If, however, by turning point you mean the point at which revolution became irresistible, that's a more difficult question to answer.
Ultimately, I'd argue that this trajectory was shaped by the experience of the French and Indian War. After the war ended, the British government took a far more active role in administering the colonies. It introduced new taxation policies in the mid-1760s and began more strongly enforcing British mercantile law. As tensions rose in the colonies, the British sent troops to North America, which only further increased tensions.
Even so, if I was to single out any one moment in this trajectory, stretching from the end of the French and Indian War until the launch of the American Revolution, I would probably choose the Intolerable Acts of 1774, passed in reaction to the Boston Tea Party.
Under the Intolerable Acts, Boston was placed under martial law, with its harbor closed until such a time as the losses from the Tea Party had been repaid. Additionally, the Intolerable Acts required that Royal Officials charged with crimes to be tried in Britain, rather than in the colonies. Meanwhile, Massachusetts saw its Charter essentially revoked. Finally, the British Government passed a new Quartering Act, allowing British officers to requisition private property for the purpose of housing their troops.
These Acts sparked outrage throughout the colonies. Indeed, be aware that the name, "Intolerable Acts" was coined within the colonies themselves. This expresses how negatively these Acts tended to be viewed from the colonial perspective. They were collectively seen as an abuse of power by which the British government was punishing an entire colony for the actions of a very small group of people. This contributed greatly to the narrative that Britain was acting more as a tyrannical force than as a legitimate government.
Additionally, note that the Intolerable Acts sparked the first meeting of the Continental Congress. The situation would only further escalate from there.