Why did the british win the battle of new york

Long Island?
Washington retreated and was outmatched and later went on to lose back to back.

Well, they outnumbered the Americans at the battle of Long Island by a good 10,000 soldiers, all in which were well trained, while the Americans were complete amateurs with little to no training at all.

At the battle of Kip's Bay (which is in Manhattan) a couple of weeks later the Americans refused to stand and fight, in which they ran away in the face of a charge by the British despite Washington's attempt to rally them.

The Americans held their ground when the British attacked at Harlem the next day, but it was a very minor and insignificant victory.

At White Plains in late October, the militia fled from the battle leaving the regulars alone.

Fort Washington, in November, was a complete disaster as the British captured 3,000 Americans due to quick planning by the American generals, and bad advice by Nathanael Greene.

I hope this answer has helped you, and I'm truly sorry if this answer has strayed from the right answer. This part in history is complicated and is very easy to mess up, just keep that in mind. :)