Last month, the bishop of Lexington, Ky., addressed hundreds of L.G.B.T. Catholics and their supporters who were meeting in Chicago at a New Ways Ministry national symposium, telling them, “Your presence and your persistence in the church is an inspiration for me and for many.”

Bishop John Stowe, O.F.M.Conv., told America that he accepted the group’s invitation because of a desire to engage in dialogue with Catholics who do not always feel welcome in the church. “Pope Francis talks about a culture of encounter, and that requires a lot of listening,” he said. “What I’ve seen among gay Catholics in my own diocese is a real desire to live their faith and the challenge to do so within a church that is not always accepting or labels them as disordered.”

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Bishop Stowe is certainly not the first bishop to address a gathering of gay and lesbian Catholics, but his insights are emblematic of recent shifts in the relationship between the church and the L.G.B.T. community. Support for pro-L.G.B.T. causes, including same-sex marriage, has risen sharply among lay Catholics in recent years.

Support for same-sex marriage among U.S. Catholics closely tracks support among the country at large.

Part of that support, several L.G.B.T. advocates said, stems from the increased visibility of gay and lesbian Americans. More people know a family member or friend who identifies as gay or lesbian and are thus more sympathetic to causes they support.

“Who am I to judge?” is one of the defining phrases of the Francis papacy, a statement praised by some Catholics. The pope has followed up that message of inclusivity with concrete actions as well,meeting with a transgender individual at the Vatican and with a gay couple during his U.S. visit in 2015.

But Brian Brown, head of the National Organization for Marriage, told America that those who see in the pope’s words and actions a softening of church teaching on marriage or sexuality “fundamentally misinterpret” his beliefs, pointing as evidence to the pope’s strong words against gender theory, the idea that male and female identity are not biologically fixed.

He notes polls that show Catholics who attend Mass weekly support same-sex marriage at lower rates than Catholics who attend Mass less frequently, yet he acknowledges that the church must do a better job overall in educating Catholics about church teaching.

“People need to know not only what the church teaches but why it teaches it,” he said. “There is a deep, deep well of anthropological, philosophical and theological beauty and truth in church teaching, and I don’t think we’ve done a good enough job in conveying that to the next generation.”

The past few decades have seen a sharp increase in the number of Americans who personally know L.G.B.T. people.

American bishops have emphasized different aspects of the church’s teaching in their outreach to L.G.B.T. Catholics and their families. Cardinal Joseph Tobin of Newark told an L.G.B.T. Catholic group that they would be “very welcome” to organize a pilgrimage to the cathedral in his city. And Cardinal Blase Cupich of Chicago, a delegate to the pope’s global meeting of bishops discussing family issues, said in 2015 that he meets with gay Catholics to understand their perspective and that gay Catholics in relationships could rely on their consciences when it comes to the question of receiving Communion. Meanwhile Archbishop Charles Chaput of Philadelphia, also a delegate to the Synod of Bishops, has said that gays and lesbians not following church teaching on chastity are not welcome to Communion.

Arthur Fitzmaurice, an Atlanta-based Catholic who through lectures and workshops advocates greater acceptance of L.G.B.T. Catholics in the church, said there have been “a lot of ups and downs” over the past decade. “Catholics in the pews generally accept people in same-sex unions and want the church to be a safe space for them, but there are still pockets in the church in areas where [L.G.B.T.] people feel isolated,” he said.

Despite shifting views among average Catholics, advocates for L.G.B.T. people say the church can still feel unwelcoming. Church teaching that homosexual acts are “intrinsically disordered” has not changed, and the Vatican reaffirmed its ban on priests with “deeply-seated homosexual tendencies” as recently as last December. Some also point to policies that prohibit gays and lesbians from holding leadership positions in parishes and to the termination of openly gay and lesbian employees from Catholic schools and other institutions.

Margie Winters was fired from her position as the head of religious education at a Philadelphia-area Catholic school in 2015 after a parent complained about her marriage to a woman. She told America that dismissals like hers lead some to drift away from the church, taking with them their creativity, energy and faith.

“We should be about the Gospel,” she said. “We should be about including people, which is what Jesus did.”

Just three months after she was fired, Ms. Winters and her wife were part of the crowd welcoming Pope Francis to the United States during a ceremony on the White House lawn, an event she described as “an incredible privilege.” She continues to be active in her parish and said she has hope that other L.G.B.T. Catholics will also feel welcome in the church.

“With any institution that is as large and has as long a history as the church, any kind of overtures are welcome,” she said. “But I also don’t expect that the church is going to change overnight. I think Pope Francis and others of his mindset are trying to change the tone and the pastoral approach to L.G.B.T. persons, but it’s a long road.”

Editor's note: In an earlier version of this story, Margie Winters was misidentified as Margie Williams. Also, in an earlier version of the chart on views toward religious exemption laws, some of the bars had incorrect lengths.