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Joshua Humphries, a shipbuilder in Philadelphia, was paid to build the first US Navy ships in 1794. Humphreys made some key improvements in their construction. Wooden ships- any ships really- tend to be subject to something called hogging and sagging. If you image a typical floating hull, the wide and bulky middle part has more flotation at the pointy ends, which means the ends tend to sink but the middle tends to rise- that's hogging. Then if a boat is between two widely spaced waves, the reverse happens- the ends tend to float in the peaks of the waves and the middle is unsupported- that's sagging. The usual construction of a big planked boat had essentially rigid circular frames of knees and ribs , with long individual planks fastened to them, and tarred rope caulking pounded between the planks. They are somewhat vulnerable to hogging and sagging, and over time it gets worse, and that flexing hull gets weaker, sails worse- and leaks more. Humphreys added long diagonal bracing, somewhat like wind braces on timber frame houses, and that made his hulls far more rigid. They could sail better, carry more guns on a single deck, and since they didn't need as many gun decks, they could sit lower in the water and be somewhat wider than usual. They were also made with good strong oak planking, from southern Live Oak. That made them harder to batter, especially the USS Constitution, and so they were quite effective. That really stood out, in a war where the US had very little success on land.

However, "better shipbuilding" is too broad. the US had only six capital ( big) ships, the British far more. And the British were able to capture two of the six. The first, the Chesapeake was taken somewhat by surprise, before the war had even been declared. It was fired upon and taken after refusing to allow a search for escaped British seamen by the HMS Leopard. The President was taken late in the War, in 1815, attempting to run past the naval blockade of New York harbor. The British were indeed surprised by how well the American boats could sail. But being able to build and expertly sail a lot of pretty good boats can be much better than having just a few very good ones.

This is a bit out of my usual area of input, but adding onto u/Bodark43 response, you really have to look at the wood supply and the strategic situation.

The British ship building yards were very technically competent and advanced in building ships. By 1812 they'd been on wartime production for roughly 30 years! They constantly tried new designs and studied captured ships for novel concepts as well. They were also quite occupied with the war on the continent, and the war with the USA was a nuisance and little more. The British would not be deploying any prime ships to North America. Further, the need to make so many ships imposed a requirement that shipbuilding drafts be something that could be executed by merchant yards and it needed to be something that could be built quickly and economically. This environment tended to discourage experimentation.

On the topic of wood, u/Bodark43 notes that the US built it ships while having access to essentially limitless supplies of hardwoods like Live Oak. Especially by 1812, but even at least 20 years before that, the demand for hardwoods for the British navy was outstripping the ability to grow new hardwoods. Robert Gardiner in "Frigates of the Napoleonic Wars" notes that "beginning in the 1790s" there was a shortage of grown knees for construction, for instance (a knee being sort of a right angle fastener bracket that traditionally trees were grown to produce). The British new oak was a superior building wood for ships and they only used softwoods in order to save oak for important constructions and high stress parts. The Royal Navy official march is "Hearts of Oak" for a reason.

To deal with the upstart Americans, the Brits laid down a round of frigates, however they didn't have enough hardwood to meet the production needs for the war in Europe in the first place, so the additional ships were made from either red or yellow pines. Even this wood was source from abroad, either from Canada or the Baltics (Gardiner seems to indicate that this building program used Canadian pine, which makes sense given that Baltic pine would have been hard to obtain in quantity given the continental system embargo Napoleon had in place). These were from the start seen as disposable ships with a short usable life due to the woods that they were made of, and in the end the ships lasted even less time than anticipated with red pine ships lasting only 4.5 years and white pine only 1 year before being forced out of service despite a designed life of 8 years before needing refit (As experienced with the Baltic pines) per Gardiner.

With the American Embargo Act of 1807 and the Non Intercourse Act of 1808, access to oak lumber from the USA would also have been greatly constricted, leaving the British with access only to domestically grown oaks (the traditional English Oak), white oak from Canada, or whatever oaks from the continent they could get smugglers to bring across the channel.

As far as the design advance that Humphrey included in the American ships, I would say that some British ship surveyors (notably Snodgrass and Seppings) had been advocating for many of the design changes that Humphrey used in the USA building program. The 1810 rebuilding of the 74 gun Tremendous saw the integration of diagonal bracing, for instance. That doesn't change the fact that Humphreys built the Constitution et al 13 years before the Tremendous refit, but by the time of the 1812 War the British had at least drawn level. This also doesn't diminish the quality of the design. The British captured the USS President in 1815 and kept it in service for 3 more years, then over 10 years after that used the same design plans to build a new one. The USS Chesapeake served as a British ship for 6 years, outlasting probably all of the pine frigates the British ordered built to fight the USA.

So, I would conclude that the comparison is difficult to make. The USA did build some very good ships for its navy, but the quality was aided by easy access to superior materials and a strategic situation that demanded quality OVER quantity (six really really good large frigates) while Britain was faced with a strategic situation that demanded as much quantity as they could find sailors and money for and still maintain just ENOUGH quality to beat any French ship or pirate, while faced with an highly constricted supply of favorable materials.