Here’s the scoop:
The practice of polygamy in the church began with Joseph Smith. It’s likely he married his first plural wife in the 1830s (Fanny Alger), but this is still considered controversial among many historians. He definitely began marrying other women in the 1840s, but he never taught the practice to the general church membership, though he did share the doctrine of polygamy with some close associates.
Then, when the Mormons were driven from Illinois and decided to leave the U.S. altogether, they settled in what is now the Salt Lake Valley in Utah. Church leaders practiced polygamy then, and eventually taught it to the general church membership. For a few decades they practiced polygamy, but (to make a long, long story short) the U.S. government eventually put pressure to stop the practice after Utah had become an official territory of the U.S. under the Treaty of Guadalupe Hidalgo with Mexico.
OK, so at this point, the U.S. tried various methods to suppress polygamy, but it was difficult because of the First Amendment right to practice one’s religion. The Poland Act, the Edmunds Act, and others were instituted to make polygamy illegal, but they never really worked until the Edmunds-Tucker Act was passed. This Act disenfranchised the Mormon Church, forced women to testify against their husbands, and allowed the government to put those under suspicion of practicing polygamy behind bars without bail or rights to habeas corpus. At this point, Wilford Woodruff, president of the church, issued what is known as “The Manifesto” where he officially ended the practice of polygamy as a teaching of the church. The U.S. allowed those Mormon families to not be broken apart, but no new plural marriages were allowed.
Some Mormons continued to marry plural wives for a short time after, and Joseph F. Smith, the church’s sixth president, issued what is known as “the Second Manifesto” wherein he said that any who entered plural marriage from that time on would be excommunicated from the church. This happened in the very early 20th century. Since then, the church has held to this standard. If you run into polygamists who call themselves Mormon, realize that if they are members of the Church of Jesus Christ of Latter-day Saints, they will be excommunicated.
So, given this history, and what today’s overwhelming majority of Mormons practice, Big Love is way, way off.