Absolutely false. The British were attempting to prosecute the war to the fullest of their ability. In fact, at the height of the war, the British deployed the majority of their army and navy to deal with the Revolution. They spent more than 250 million pounds. It was, at the time, the largest war Britain had fought in terms of men and material. (It would be dwarfed by the Napoleonic Wars only a few decades later though.)

It's true that, prior to 1775, the British did not act to crush the arguably illegal organizations and committees formed by the Revolutionaries. But the British did not consider the colonies to be in open revolt at that point. A military response was considered provocative. This was especially true of the military on the ground. It's now known that the local British commanders were actually conspicuously trying not to provoke the colonists, even to the point of loosely interpreting orders so as to not provoke an uprising. This is not to overstate their sympathy: they were trying to disarm the colonists and reduce them to a state where they could not effectively resist British military power. But this was because, at the time, they thought a political solution would resolve the crisis.

Once the war broke out, the British treated it like they had treated previous civil wars. They were conspicuously trying to avoid being destructive, but this had been a feature of English civil wars going back to at least the War of the Roses. As one veteran reflected on the English Civil War, it was in no everyone's interest to preserve the country so the country remained preserved. (This is not to say there was no destruction, but it was less than might have been expected in an enemy nation during the Seven Years War or Napoleonic War). But this again wasn't 'not trying'. It had a fairly basic logic. Firstly, these people would once again be British subjects upon victory and should not be embittered (to say nothing of those still loyal). Secondly, if the colonies were laid low then they were less valuable, and the British hoped to continue to profit from holding them.

But they were trying to win, right down to political purges, attempts to destroy hostile political structures, executions and imprisonment, and, of course, deploying an incredibly large amount of men and material to accomplish this aim.

Why didn't Britain win then? Well, conditions favored the defenders. Holding the large hostile country was basically impossible. The US was more than 400,000 square miles and had a population a little below three million people. Virtually all of that population had a gun and some military experience. And on top of the native logistics problems, British troops had to be shipped thousands of miles, usually from Europe. And then, on top of that, once British forces were tied up fighting the rebels, everyone with a score to settle dogpiled Britain. Everyone from France to Indian Princes declared war on the British to take advantage of British weakness. But these are not problems about lack of will.

And this is a theme of early Anglo-American wars. The British usually had the advantage over Americans in conventional warfare. But the British could not effectively leverage these victories and were often distracted by opponents in other theaters. The Americans, meanwhile, lost more often than they won but with every victory pushed the British back or out. And they only had one theater: their homeland. The famous 'rocket's red glare' line in the American National Anthem effectively shows this dynamic during the War of 1812. It refers to the Americans holding a single fort at the mouth of a somewhat important river after Washington DC had been taken. The British could overrun the major city of the area and many of its forts and greatest fortifications. But because the British could not drive all American forces out of the area everywhere, the Americans could successfully mass for an overwhelming counterattack and the British decided to withdraw rather than face them.

This was true at the time too. The British took the capital of the US, most of its major cities at one time or another, and inflicted defeats on virtually every army the US had at one point or another. But they were not able to capitalize on this to create a strong hold on American territory or destroy American armies. The US armies usually lived to fight another day and, as soon as the British armies suffered a defeat, the Americans often pushed them far back.

So, in summary, the Americans had a massive advantage as defenders in a distant place, benefitted from allies and other people opportunistically attacking Britain, and the fact the lead up to the war was a slow build up of tensions gave the Americans the ability to organize when most vulnerable.