A census of the Los Angeles area was taken in 1836 that counted all non-Indian residents. It put the population somewhere below 2,000,^1 with the American population being about 20.^2

There were population counts of San Francisco in this era as well, and the non-Indian population peaked at 373 in 1815, while it was about 160 in 1842.^3 Most of these people would have been Mexican.

When the Mexican-American War broke out, the total non-Indian population of California was somewhere in the low thousands, certainly somewhere below 10,000.

It was during the Mexican-American War that the non-Indian population increased significantly. That war was fought beginning in April 1846, and ended in February 1848, with the U.S. gaining possession of "the West". California became a U.S. state in 1850, and it was counted that year in the U.S. Census, with about 90,000 non-Indian residents. San Francisco is not included in this total, but a 1852 census of San Francisco does survive, and the total of the whole state in 1850 is estimate to be somewhere around 120,000. Again, many of these would have been Mexican, though by then, the majority were probably American.

The Gold Rush started in 1848, and most of these people came during that era, motivated by the Gold Rush or some other reason, such as homesteading, under the 1841 Distribution Pre-Emption Act.^4

This is the reason that the "Wild West" is associated with the Gold Rush and after. The first big push of a non-Indian English-speaking population didn't occur until after the end of the Mexican-American War in 1848, which also marks the start of the Gold Rush. Transportation was another barrier. The federal government granted 2 million acres of land to the Illinois Central Railroad to build a railroad out west in 1852,^4 though there was no completed East-West link until the driving of the Golden Spike that connected the Central Pacific and Union Pacific railroads on May 10, 1869. The 1860 presidential election featured planks on both the Republican^5 and the Northern Democratic^6 Parties' platforms about finishing the transcontinental railroad.

If your question is about gunslingers, duels, brothels, drinking, Indian wars, lawlessness, and the honorable hero, while some of this has a basis in history, most of the popular conception of the "Wild West" can be attributed to the "Western" genre of movies and literature. So you might blame their popularity on the popular conception of the "Wild West," starting before 1910, with films such as The Great Train Roberry (1903), The Story of the Kelly Gang (1906) (though that one is actually about the "Wild West" of Australia), and the films of William N. Selig, the head of one of the "Edison Trust" studios who was the first to establish a permanent studio in L.A., in part for the express purpose of being able to more easily film Westerns. Obviously, the "Western" film genre got more salacious as time went on, so I'll leave it to someone else to explain how its tropes came to be in post-1900 era, but for this answer, just know that the Wild West's associations with outlaw behavior were part of American pop culture before 1910.

But why hadn't these tropes been associated with the frontier settlements before the "Wild West" entered the picture? Well, they kind of had. They were just usurped by the tales of the "Wild West".

One of the most popular American fiction writers of the first half of the 19th Century was James Fenimore Cooper, who is mostly remembered now for writing The Last of the Mohicans. But that book is actually one of a series referred to as the "Leatherstocking Tales," all adventure books set in the late 1700s and early 1800s on the frontier in New York, the Midwest, and Kansas. The books feature battles between "pioneers" and Indians. The frontier heroes build log cabins. They struggle through adversity for a better life for their family. Unjustified murders force the frontier hero to seek justice in some way, though all he wants is peace. There are challenges to the hero's conviction, and in at least one instance, a duel. There are damsels in distress. The main protaganist, Natty "The Deerslayer" Bumpo, is an expert shooter, and one of the books has him arguing with another character about who killed a buck--an argument over honor. Some hard drinking is portrayed, such as during Christmas scenes in The Pioneers,^7 though it's usually the Indians who are portrayed as drunk, while the hero remains sober. They don't always feature the more salacious aspects of later Westerns, but a lot of the "Wild West" tropes are already there.

The "Leatherstocking Tales" were all published between 1823 and 1841, which was the same time as some of the other frontier mythology got its start, probably most popularly with the tales of Daniel Boone (starting with the 1833 publication of myth-filled Biographical Memoir of Daniel Boone, the First Settler of Kentucky) and of Davy Crockett (starting with the 1834 publication of myth-filled A Narrative of the Life of David Crockett, Written by Himself).

These were far from all. It was an important American genre of literature--perhaps the first genre of literature invented in America. In The Frontier in American Literature by Edwin S. Fussell, the author discusses several other important works, arguing that the genre dates at least as far back as the 1782 book Letters of An American Farmer.^8 Fussell also discusses how the concept of "the West" expanded over time, quoting James Fenimore Cooper's 1854 introduction to his book The Prairie, where Cooper says that, when the book was first written in 1820, the "Far West" meant Missouri, but by 1854, it meant California.^8

As for the reality behind it all, well, the Wild West tropes of manhood, dueling, hard drinking, and brothels may not necessarily be "exaggerated", but probably over-emphasized, because all of these aspects of American life were there wherever there had been a frontier--and after. The early historical documents of New England, Virginia, and New York all discuss taverns, public drunkenness, and lawlessness. All of them had court cases involving prostitution. This continued on during the first half of the 1800s. As just one example, an item in The Literary Magazine of New York talked about the importance of newspapers in Ohio, "which afford [the people there] entertainment which might, perhaps, be sought elsewhere, in the theater, the porterhouse, or the brothel."^9

Duels, of course, were also prevalent in reality out East. The most famous is probably the Burr/Hamilton duel, but Andrew Jackson, Henry Clay, Sam Houston, and even Abraham Lincoln were all challenged to a duel in their time (individual duels--they weren't involved in some time-traveling battle royale).

And the search for a better life was certainly a motivation for many settlers to relocate to the frontier, for cheap or sometimes even free land. As Fussell argues in his book, many of the early entries in the "frontier" genre of literature amounted to not much than propaganda pieces encouraging settlement beyond the East Coast, painting a rosy picture based on manhood and a "better life" when, in reality, the frontier was a much more difficult, and not often more rewarding, life than what was available out East.

TL;DR: The "Wild West" is mostly associated with the era of the Gold Rush and after because that's when the U.S. first gained possession of the land west of the Louisiana Purchase, and that's when Americans started to settle out there. Before that, a lot of the "Wild West" tropes were already associated with the "frontier" genre of literature, as exemplified by James Fenimore Cooper's "Leatherstocking Tales," and the tales of Daniel Boone and Davy Crockett. More historically, a lot of the tropes did have a basis in reality wherever there was a frontier in America. It's just that the location of "the West" changed over time, so the last "West," the "Wild West" of California/the Mountain West is what it's now associated with, due to pop fiction. Occasionally, though, near-West portrayals are still found: The Little House on the Prairie books and TV series, Drums Along the Mohawk starring Henry Fonda, The Revenant starring Leonardo DiCaprio, etc.

I'm no expert on the Wild West genre itself, so if your question is specifically about how Wild West cliches developed over time in popular fiction after 1900, maybe someone else can fill you in.

SOURCES:

1. History of California: 1825-1840 by Hubert Howe Bancroft

2. This Small City Will be a Mexican Paradise: Exploring the Origins of Mexican Culture in Los Angeles, 1821-1846 by Michael J. González and Mike Gonzalez

3. The Colonial History of the City of San Francisco by John Whipple Dwinelle

4. Transforming California: A Political History of Land Use and Development by Stephanie S. Pincetl

5. "Republican Party Platforms, Republican Party Platform of 1860 Online" by Gerhard Peters and John T. Woolley, The American Presidency Project

6. "Democratic Party Platform; June 18, 1860", Yale Law School Avalon Project

7. The Pioneers by James Fenimore Cooper

8. Frontier in American Literature by Edwin S. Fussell

9. "Cheap Postage," The Literary American, Jan 6, 1849

EDIT: Thanks for the gold!