It's true that the British Army and Navy were the best in the world at the time, and the German mercenaries the best troops in Europe. But the Americans had unique advantages. The technology of weaponry meant that the average person had a firearm the equal of a professional soldier's, and communities often bought a cannon for the local militia, so the population was well armed to start with. Where the British were schooled in European warfare, based on armies en masse manuvuering around fixed pivots of arsenals or fortresses, the Americans were used to fighting in the manner of Native Americans. This gave them an early advantage, as did the high proportion of farmers in the Continental Army. British and Hessian officers often remarked on the rapidity with which Americans built earthworks.
The Army commands were opposites. The British were thoroughly professional, and included both British and German officers of great ability. The British troops, on the conventional battlefield, were the best in the world. But Washington consistently avoided disaster in the battles he lost, knew when to gamble and when to escape in the aftermath of battles he won, and with the help of von Stueben eventually built up regular line units which could stand up to the British in major battles.
The early British plan for cutting the colonies in half through the Hudson and the later plan of cutting off the deep South from the North were conventional but well-laid plans. But the unconventional tactics of Benedict Arnold (fighting for America) in the North and Nathaniel Greene and William Washington in the South spoiled those plans. The British were well-schooled and experienced, but their strategies were unrealistic in the light of conditions in America. They consistently underestimated the difficulty of terrain and supply, and while their tactics and training were superior at first, that changed as time passed.
The British Navy was the best in the world, but American sailors were the best smugglers, and the French and Spanish were willing to give money in munitions and extend credit to the colonial government. The American Navy was small, but American privateers ran rampant throughout the Carribean and Atlantic, and even in the English Channel.
Many in Britian were in sympathy with the colonials, especially among the 50% of the population which was poor, many of whom had relatives in the colonies. The merchants were also opposed to the war, and as losses in shipping to privateers mounted their opposition increased. The economic dislocation led to higher unemployment, and eventually the cost of the war, especially after it expanded to include the French in actual hostilities, threatened to become far more expensive than the British government could hope to recoup after the war, even if they won. The most serious downside for the British, in the end, was the disaffection of a large part of the public and a split within both Parliament and the military over the entirety of colonial policy.