The Puritan Pilgrims, in particular, came to America when King Charles I of England began persecuting Puritans. Many other Separatist groups fled to the Netherlands.
IIRC, The pilgrims went to Holland to avoid persecution, they later went to America because it offered more opportunities and they were losing too many people to the “corruption” of the cities.
Yes, many Separatists did go to Holland to avoid persecution, as England had nationalized religion to the point of making it illegal for people in England not to attend the Church of England. But many of these Separatists later returned to England as the Netherlands was not as ideal a place for them as they had hoped. It was from England that the famous Mayflower ship departed with the Puritans on board.
By the way, "pilgrim" was not their religion -- that was merely a term applied to anyone taking a journey for a religious purpose. Many Americans, however, view the term "pilgrim" as exclusively referring to those early Separatists who came here. A lot of Americans also think that the "pilgrims" got along great with the indigenous population already here, and that our holiday of Thanksgiving represents a charming celebration of that getting-along. The "First Thanksgiving" may have turned out that way, but it is a fact that Governor Bradford of Plymouth Colony proclaimed a Thanksgiving feast following the war with the indigenous Pequot tribe, in celebration of "the bloody victory, thanking God that the battle had been won."
However, if one teaches this history to children in our schools these days, one is likely to initiate another kind of war, with angry parents screaming about CRT or whatever else makes them feel bad to hear. Maybe, when we have a national religion of Christianity in this country, we won't need to worry about history that contradicts our beloved legends? Nah.