The details were fairly commonly known - there were enough survivors with diverse enough experiences (crew, passengers, etc.) that the events were able to be reconstructed. While the first newspaper reports were depressingly and misleadingly optimistic ...

World's Biggest Ship Reported in Bad Shape After Collision at Night and Now Being Towed to Halifax by the Allen Liner Virginia

... From another source it was reported to the [White Star] line late this afternoon that all the disabled vessel's passengers had been put aboard the Carpathia, Parisian and Virginian.

(The Evening World, 4/15/1912)

... the interviews and inquests soon produced a somewhat coherent story. The Titanic was moving at a good clip at night, struck an iceberg, ripping open the side and filling too many of the "watertight" compartments, and began to sink as the band played "Nearer, My God, To Thee"; the passengers had little idea of what was happening, many disbelieving that the ship could be in any distress, so too few got in the lifeboats. Stories of chivalrous behavior by individual first-class male passengers and crewmen were common, but so were rumors about the men who survived (such as Joseph Bruce Ismay and Cosmo Duff Gordon) dressing as women or buying their way onto lifeboats. All of the pieces of the Titanic mythos were in place at that time: heroism in the actions of men who put their wives into lifeboats when they knew they would die, the strengthening of gender roles, the class issues in general (upper-class nobility, lower-class men in the crew helping upper-class women, steerage passengers being abandoned, etc.), the religious comfort of the hymn, the hubris of the builders and directors, and, less an overt part of the story today, issues of nationality and race (to some extent tied in with the class issues, but also relevant to the US vs UK rivalry).

The story's relevance to pop culture waxed and waned. While several films (and many books) came out right away - one, about the story of one passenger's survival, came out within a month of the sinking - the visceral, emotional nature of the story faded, and it became a cautionary tale and an example that led to changes in ship safety. It kind of skated under the radar until 1929, when the play The Berg premiered and was turned into the movie Atlantic; both (especially the latter, after complaints from the White Star Line) made an effort to avoid any specific names that would point to the Titanic, but, I mean, they were about a luxury liner leaving England that hit an iceberg and sank tragically. While both involved an ensemble cast of diverse characters, something that would be fairly common in depictions of the tragedy, the central storyline involved a priest finding his calling again and providing spiritual aid to the male main characters (plus one woman) who chose to go down with the ship - the focus on the heroism of the victims, something that had been a constant since the first reports in the newspaper. The next English-language film, Titanic, didn't come out until 1953. Focusing on an estranged, wealthy American couple and their children, it likewise ended by splitting up the family (even to the point of having the ten-year-old son go down on the ship with his father). The liner and its tragedy were used again as a tool for making thematic points: the family's troubles are all about European artificiality vs. American genuineness, extravagance vs. "real" middle-class living, father vs. mother; the ending emphasizes that the genuine, middle-class mother has been mostly correct, but that the gender roles upended in their disagreement must be fixed.

It's not until 1958's A Night to Remember that we really get the Titanic story for Titanic's sake. While the film has some fictional characters, it also tried to depict the sinking in a semi-documentary way, showing a number of fictionalized versions of crew members and known passengers and focusing on the how and why rather than a story. (It was based on a non-fiction book of the same name from 1955 which took basically the same viewpoint.) For the first time, there was more emphasis being put on the great tragedy of the overwhelming loss of life in a terrifying and drawn-out accident, the ways it all could have been avoided through human actions, and the class inequality relating to who got the opportunity of escape - issues that are certainly at the forefront of how we perceive the disaster today. The next Titanic-sinking story on film came out after the exploration of the wreck itself - James Cameron's 1997 Titanic. Although Jack, Rose, their romance, and the ancillary characters give much more of a personal dramatic storyline than A Night to Remember had, those themes carried through. My impression is that between these two films, apart from the time of the rediscovery, there was comparatively little pop-cultural interest in the ship, with the real outpouring of fiction (particularly YA/juvenile fiction) coming after the 1997 film.

As far as conspiracy theories go ... there are a few alternative ideas as to how the sinking went, largely late-twentieth-century and largely just about technical details. The real conspiracy theory is the one put forth by Robin Gardiner just a few years ago stating that the boat that sank was actually the Olympic, a sister ship that resembled the Titanic and had been somewhat damaged a year earlier, and that the Olympic that sailed after the "Titanic" went down for the insurance money was actually the Titanic. There don't seem to have been any widespread conspiracy theories in the early twentieth century that would be comparable to "jet fuel can't melt steel beams", perhaps because the details were so thoroughly hashed out at the time.

Recommended reading:

The Titanic in Myth and Memory: Representations in Visual and Literary Culture, edited by Tim Bergfelder and Sarah Street (2004)

The Myth of the Titanic, by Richard Howells (1999)