I've often seen people blame the "Puritan roots" of the U.S. as the reason for a lot of the more conservative views that are predominant today. Probably the most common is the more conservative views on sex, especially how protective the U.S. is of letting teens be exposed to sexually explicit content or even non-sexual nudity. But I've also seen to attributed to other things, such as simply the lower rate of people who identify as atheist compared to many European countries.
I've always been skeptical of this explanation, though, so I wanted to see what everyone here thought. As I understand it, Puritanism wasn't a movement about purity in terms of abstaining from sex, alcohol, etc., it was about "purifying" the Anglican church of many of the practices they kept from Catholicism that didn't have any Biblical basis. Although it seems that a lot of the Puritan communities that emigrated to the U.S. were quite socially conservative nonetheless, so maybe there is some truth to it.
You are right about Puritanism specifically being about "purifying" the Church of Catholic influences. It's also worth nothing that most of the Puritans who immigrated to what is now the US (more actually immigrated to other places like the West Indies and the Netherlands, but that's another topic) settled in what is now New England, with a few elsewhere, but largely absent from places like the Southern Colonies or New York. What was Puritanism through various twists and turns evolved into Congregationalism and Unitarianism, which are two of the most socially and politically liberal Protestant sects in the US today (although "Banned in Boston" was still a thing long into the 20th century).
Another point is that it's really hard, especially when talking about cultural practices or outlooks, to talk about "Europe" as a single entity, or even the United States as one. Las Vegas and rural Utah are right next to each other, but have vastly different cultures and conceptions of public morality. Likewise Amsterdam and rural Moldova are both "Europe", but might as well be different continents in terms of morality and religion. And often these broad brush-strokes obscure more than they illuminate: same sex marriage, for example, is legal in all of the United States, but not even in all of Western Europe (and is outright banned in many Eastern European countries). As under attack as it is, abortion is still legal in all of the United States, but is not (yet) legal in Ireland.
One of the reasons that there is this perceived social and moral conservatism in the United States contrasted with a more liberal public morality in Western Europe (at least) probably has to do with how politics and issues of public morality intersected in both regions in the 1960s-1980s. Starting in the 1960s, a number of Western European countries such as the UK began to liberalize laws relating to public morality, such as the Sexual Offences Act of 1967 (decriminalizing sex between consenting men over 21) and the Abortion Act of 1967, legalizing abortion.
In contrast, religious leaders in the United States such as Jerry Falwell (a Southern Baptist) and James Dobson (from the evangelical Church of the Nazarene) began to eschew their Protestant sects' traditions of separation from politics*, and organize on the grassroots and national level to have a greater influence in promoting legislation on public morality. These groups came together especially in the 1970s (Falwell organizing the Moral Majority, Dobson organizing Focus on the Family), and were bringing together discontent at and resistance to a number of trends: the legalization of abortion with Roe v. Wade in 1973, increasing public acceptance of sexual topics, sexual discussion and sexuality in the 1960s-1970s, the Supreme Court banning prayer in public schools in 1962, and desegregation (especially of schools) being large motivators, especially in the American South.
Despite an evangelical Southern Christian being elected president in 1976 in the form of Jimmy Carter, these organized political movements found the greatest traction in the Republican Party, starting with the Moral Majority's endorsement of (divorced and not very religious) Reagan for president in 1980. This was also a period where the Republican Party's "Southern Strategy" was bearing fruit, with the party rallying white former Democrats disenchanted with the Democratic Party's stance on desegregation to the Republican fold, in part using calls for conservative moral issues in place of explicit opposition to desegregation (this is when Strom Thurmond and Jesse Helms switched parties, for example). It's mostly from this affiliation with Republican Presidents and Senators at the national level and having increasing control of state level politics in the South and Midwest that a political conservatism with a heavy focus on public morality and having strong connections to various flavors of Protestant Christianity developed into a phenomenon.
*Someone like Billy Graham is an example of a public evangelist who both maintained political influence, but also trained to remain officially bipartisan,.
Source:
Mostly James T. Patterson's Restless Giant: The United States from Watergate to Bush v. Gore, and he has a good list of further references on the subject in his bibliographical essay at the end of the book which I recommend.