There are several reasons for this. I unfortunately can't go into a great amount of detail about military doctrine and how it changed from 1776 to 1789-1815, but I'll make a few general points.

One of the first things that always has to be stressed is this: The British Empire's main military asset was always the Royal Navy. While the Navy was probably the single most powerful naval force in the world at the time, the British Army was somewhere in the ballpark of 40,000 - 50,000 men in size in the early 1770s. This force was spread throughout the Empire, not concentrated in the Isles or elsewhere. Britain had spent decades fighting costly wars in Europe and overseas (Seven Years' War, War of the Austrian Succession, etc.) and relied primarily on diplomatic maneuvering with continental European powers to compensate for its relatively small army. Prussia, for example, a state with a fraction of the population of the British Empire at the time, had an army of roughly 80,000 men when it invaded Silesia in the 1740s. By negotiating alliances with some combination of Prussia, Austria, Spain, etc. in large conflicts, Britain could focus on dominating the seas while allowing its allies to commit their armies. This strategy worked for them in the past, and correct me if I'm wrong but they employed a similar strategy in the Napoleonic Wars as well.

Strategically, this makes sense. The Royal Navy allowed the British to project their strength across continents and protect trade, fueling the British economy and cementing their status as a commercial maritime power. If the navy could ship some troops and supplies to India and British diplomacy and money could sway local rulers to provide assistance during war, there was little need to maintain a huge standing army. This is especially true given the monumental financial burden of maintain large standing armies and professional navies during the 18th century. France and Austria in particular come to mind when thinking of the financial difficulties of maintaining a military at the time. If Britain could avoid the financial strain of maintaining a large army by compensating with naval power and diplomacy, it made sense to do so. We saw this even in the Revolution itself: Britain relied on German mercenaries and diplomatic maneuvering to prevent external intervention.

Now, moving on to what may have happened had North responded differently. First things first, let's make a quick point: Given the lack of regulars, ships, and experienced officers fighting for the patriots, it's easy to see why the British government may have underestimated the seriousness of the rebellion (at least initially). Although a lack of "peacetime" spending of the British army left it in a much-to-be-desired state come 1776, the colonies were in even worse shape militarily. There's a tendency to view a patriot victory as inevitable, and there's some merit in that: Controling a 2+ million person geographically expansive rebellion across an ocean is difficult. But without French powder and arms, French and Spanish ships, Polish and German officers, etc., things may have turned out differently. That being said, if Britain had committed what amounted to more than the entirety of its army at the beginning of the revolt, it could have worked. The British were looking for a decisive victory against the rebels that would bring them to the negotiating table, where they would use their expertise to undermine support for the rebellion in the upper echelons of society and gradually bring the colonies back into line. It wouldn't have been impossible, ~50% of the colonists remained loyal to the crown.

However, at the same time, amassing an army of that size in the Americas would have come at such a great financial, strategic, and logistical cost that the risks may have our weighted the benefits. Britain was preoccupied with India and had to account for the defense of the Caribbean, Canada, and the Isles. Bankrupting an empire and leaving its heartland and the majority of its overseas possessions vulnerable to crush a rebellion that initially wasn't seen as highly threatening simply doesn't make sense. The Thirteen Colonies were important, but they weren't important enough to warrant going nuclear on rebellious colonies that had a history of division and no standing army. Once the colonies, seemingly against all odds, got their act together the British did commit many more men to the conflict. Remember how the British army was only about 40,000 - 50,000 men shortly before the outbreak of the war? Between mercenaries, loyalists, and a massive spike in recruitment after the outbreak of the war, the British committed somewhere around 70,000 - 90,000 troops/allies to the war in North America and several thousand at Gibraltar. It was just too little too late.

Final point, I'm assuming by "army sizes" you mean the size of individual armies engaged in pitched battles. The strategic approach to the war in the colonies was very different to that in Europe during the French Revolution/Napoleonic Wars. British and colonial troops were spread across a large geographic area with key forts and cities spread across the region. The war was for control of this land, so Britain had to retain control of as many of these forts and cities as possible. The whole war didn't revolve around New York, New Jersey, and Pennsylvania, there was significant fighting in the south as well. Given the size of the militaries and the geographic size of the war at the time, having ~10,000 (give or take) men on each side at decisive battles and several thousand less at less notable engagements seems reasonable.