I am not a military historian per se, and someone who has a better background in tactical military history may say something slightly different, but the turning point then, and now, is often seen as the Battles of Saratoga in upstate New York that occurred in September and October 1777, which resulted in an American victory. Led by General John Burgoyne, the British strategy was to split the northern colonies in two by taking control of the length of the Hudson River, which would cut off New England from New Jersey, Pennsylvania, and all points south, disrupt the supply lines, and enable the British to attack smaller, divided fighting forces of Americans. It was a "divide and conquer" strategy. The British already controlled New York City at the mouth of the Hudson, and they were attempting to take control of the river from the north. Saratoga isn't far north of Schenectady, Troy, and Albany, and once the British controlled Albany, then they could surround the Patriots from both sides until all of the Hudson River Valley was theirs, which would cut off all realistic routes between Boston and Philadelphia.

But the Americans prevailed, and beat back the British, and the British were never again able to close in on Albany and the Patriot strongholds in the Hudson River Valley, including Kingston, Poughkeepsie, and West Point. This didn't only have importance for the Americans militarily, but it had importance for them politically (France offered assistance to the Americans after this victory) and psychologically as well. The Revolution after the Declaration of Independence had started out with a series of losses for the Americans, including at the Battle of Brooklyn and the Battle of White Plains, though the American victory at the Battle of Trenton with Washington's crossing of the Delaware offered early encouragement that the Americans weren't destined to lose and lose quickly. But after the Battles of Saratoga, the Patriot side actually started to feel that they could actually outright win.

To that point, in a lot of veterans' pension applications and other such contemporary records, the Battles of Saratoga are often referred to as "Burgoyne's Defeat" rather than by the names of the battles themselves. The event held a special place in the hearts of many Americans who lived through the Revolution as the turning point of the war. One example is in the 1836 obituary of William White, published in the Alexandria (Va.) Gazette. White was the Presiding Bishop of the American Episcopal Church--the U.S. successor to the Anglican Church after it split from that church's authority with the monarch as its head:

"[William White] was for some time Chaplain to Congress, the duties of which he performed to the entire satisfaction of that body. He accepted the office after the British forces entered Philadelphia, and before intelligence was received of Burgoyne's defeat. On the removal of the Federal Congress from N. York to Philadelphia he became one of the two Chaplains to that body, and continued to officiate in that capacity till the removal of the Government to Washington in 1801."

More contemporary, the battles' importance is reflected in the titles of the books Saratoga: Turning Point of America's Revolutionary War by Robert M. Ketchum (1997), Saratoga: A Military History of the Decisive Campaign of the American Revolution by John F. Luzader (2008), and Saratoga 1777: Turning Point of a Revolution by Brendan Morrissey and Adam Hook (1961). Other histories of the battles include March To Saratoga: General Burgoyne And The American Campaign, 1777 by Harrison Bird (1963) and The Generals of Saratoga: John Burgoyne and Horatio Gates by Max Mintz (1990).

If you want to understand how the battles fit in with the greater military history of the Revolution, you might want to try The American Revolution in Its Political and Military Aspects, 1763-1783 by Eric Robson (1966), or Almost a Miracle: The American Victory in the War of Independence by John Ferling (2007).

Another, more compact overview of the Revolution can be found in The American Revolution: A History by Gordon S. Wood (2002), who puts the battles' importance succinctly:

"Saratoga was the turning point. It suggested that reconquest of America might be beyond British strength. It brought France openly into the struggle. And it led to a change in the British command and a fundamental alteration in strategy."

That all said, it's a much-studied war with much written about it, so you can probably find literature on it that points to other events and victories throughout the war that helped to turn the tide. Others might chime in with such information, but suffice it to say that the Battles of Saratoga are often pointed to as the turning point of the war by contemporary historians, and by the Americans who fought in and lived through the American Revolution.