While accepting the spirit in which it was asked, I would dispute the premise of your question. Like any major political issue, the American relationship to immigration has changed considerably over its history; on September 13, 2002 (i.e., 20 years ago) it was much harder to immigrate to the US than it was in, say, 1802. But it was much easier to immigrate in 2002 than in, say, 1932.

But to the extent that you could identify a “switch” in American policy, it would stem from the flows of immigrants in the latter half of the 19th century that are so famous in American popular consciousness. From the 1870s-1900, you see approximately 11 million people immigrate to the US, more people than had immigrated to the US than in the 17th, 18th, and first part of the 19th century combined. From 1900-1915, another 12 million would immigrate to the US. This influx of immigration prompted, for the first time, federal interest in immigration regulation. Prior to this, there was no federal regulation of immigration at all. But in the 1880s Congress began to pass general immigration laws, including the famous Chinese Exclusion Act as well as the first general federal immigration law. By 1891, the federal government took exclusive control over immigration from the states (opening up the famous Ellis Island in the process---Ellis Island is in fact one of the first steps the federal government took towards restricting immigration).

From there you have an approach to immigration that changes from administration to administration, and the ebbs and flows of power among the competing social forces that are pro- and anti-immigrant---a complex arrangement of labor, capital, nativist, anti-communist, and ethnic interests, all pulling in different directions. I won’t go into details on the many laws and policy changes in the first 25 years of the 20th century, but by 1924 you see the Reed-Johnson Act, which is arguably the high water mark of immigration restriction in the US. This Act banned immigration from Asia, sharply limited total immigration from non-European countries, and set national quotas on immigrants based on that country’s share of the population in the 1890 census (and you’ll note, this date was chosen to precede the huge influx of immigrants---many of them Southern or Eastern Europeans---that arrived in the huge waves from 1890-1915 that I noted above). This restriction continued for the next few decades, sharply limiting immigration from anywhere except Western Europe.

But this also wasn’t the end of immigration policy in the US---by 1952 these quota restrictions were eased somewhat, and in the Immigration and Nationality Act of 1965, the quota system was eliminated, and replaced by a more flexible overall hemispherical cap with flat nationwide caps.

But the really foundational change in the 1965 Act was that it allowed family reunification outside the set numerical caps. What this means is that, if one immigrant came over, that immigrant could eventually bring over their parents, children, and siblings, outside the general cap system. Once those people can immigrate, they can bring their family members over. It’s hard to overstate the fundamental change this policy has had on the ethnic makeup of the United States---Asians, for example, make up a quarter of all immigrants to the US since 1965 (up from none!). Most immigrants to the US now come via family reunification. This system has remained largely in place, with one major addition: the creation of the Diversity Visa (aka the “green card lottery”) in 1990, which was implemented for the express purpose of increasing the diversity of the US’s immigrant population (with the same multiplier effects from family reunification over time).

Thus I would say it's not accurate to say there was a single switch in American immigration policy---while we're obviously no longer in an era of no regulation, the post 1965- and 1990-era is arguably the most liberal era of immigration policy since the federal government began regulating it, with dramatic implications for the ethnic and cultural diversity of the country (not to say that there aren't nevertheless many restrictions, problems, and flaws in the system).

The above is a very general overview of the history of American immigration policy. Some of my sources are very dry legal textbooks, but if you’d like to know more, I would recommend Roger Daniels’s Guarding the Golden Door or his Coming to America, or Aristide Zolberg’s A Nation by Design, all very good histories of US immigration policy (the latter is my favorite, the first is probably the most accessible).

PS: Some other side points:

(1) Generally speaking I’m skeptical of drawing close comparisons to Europe and the US in terms of immigration policy---the US has a fundamentally different relationship with immigration than Europe, and has historically been more open to both immigrants and asylum seekers than Europe has. A better grouping would be the Western Hemisphere vs. Eastern Hemisphere; Argentina, to take just one example, was almost as popular a destination as the US for European immigrants in the late 19th century.

(2) Re: internment of Japanese citizens during WWII: the internment of foreign nationals actually has its roots in one of the oldest federal laws: the 1798 Alien Enemies Act, which permits the internment and removal of noncitizens from countries with which the US is at war; it has been used in the War of 1812, WWI, WWII, and is still good law today. What was new in WWII was the internment of Americans of Japanese descent.