You can have cholera, plague, or a very virulent influenza strike in Europe. In 1780 the state of medical science was such that nothing could be done. While sanitation and cleanliness was "better" than the Middle Ages, especially if the plague was primarily pneumonic it could be devastating. One of the factors in spread of a pandemic is ease of communications/transportation. While RRs were in the future, the roads, canal systems, and seaborne trade were all much better and more rapid than during the Middle Ages.
It is entirely possible the pandemic could jump the Atlantic, but given the much lower population density in the USA/North America at that time, and the larger distances and poorer roads, spread would be limited.
To get a pandemic you need a combination of population density (of a non-immune population), and adequate transportation to spread it beyond initial area of outbreak. A sick individual has to get from point A to point B when infected at point A but still contagious (or not yet dead) when arriving at point B.
One possible effect of a European pandemic, especially if the death toll (as a percentage) in the USA is much lower, is that you could see more rapid industrial development in the USA with a reduced European output. Also, depending upon which parts of Europe are hit the hardest, and also if there are any reactions against religious or other minorities, there will be a shift in who immigrates to America, as well as probably reduced numbers. Huge butterflies.