Why did so many immigrants come to the United States during the nineteenth century?

In the 1800s Farmers had large families. The children grew up helping on the farm. The farm supported one family. If the family had several sons, one might stay home and remain a farmer. He would inherit the farm. The remaining sons would need to find other work. They might be able to find other farms or they would need to go to cities. After 1890 in the US, the Western Frontier no longer existed. Excess farm sons would no longer go west and get a new farm. Likewise only one daughter could marry a farmer. The daughters would need to go elsewhere.

So when there were 10 children living on the farm, 5 sons and 5 daughters, only 2 could remain living on local farms. The other 8 would need to go somewhere else.

Few jobs existed in the small towns. Like always, the farmers' children went to the cities where they thought the jobs were.