I actually have a previous answer on this very topic! I shall paste it below:

The Titanic was not really touted as unsinkable! Or, at least, this wasn't the major selling point that it's presented as today.

Immediately after the sinking, the Titanic mythos began to develop, as I discussed in this recent answer. The idea that "Nearer My God To Thee" was playing as it sank, for instance, isn't true, but became indelibly attached to the event. There are also aspects of the mythos that are not necessarily myths, but are part of what comes to mind when you think of the Titanic's sinking, the narrative that everyone knows and is included in every work of fiction related to it: women and children first, or steerage passengers being locked below decks. The engineers and builders being arrogant and hubristic, calling the ship "unsinkable", is part of this mythos.

At the beginning of the extended flashback in the film Titanic, Rose's mother says, "So this is the ship they say is unsinkable," as though the public discourse is abuzz on the topic. The 1955 book A Night to Remember set the idea that everyone smugly considered it unsinkable, and publications of the following decades would popularize lines like "God himself could not sink this ship!" and the belief that it was "safer than a lifeboat". All of this gives the impression that ships were constantly sinking and transportation companies and potential passengers were desperate for relief - which was not the case. On the other hand, a rival view (we could call it a second-option bias) appeared in the 1980s that said nobody had ever called Titanic "unsinkable" until after it sank; fewer people heard of this, but it became the insider view for a time.

As stated in the earlier answer, the idea did come up before the sinking - around the time that Titanic and its sister ship, Olympic, were built, both ships' unsinkability was mentioned. There's a 1910 promotional leaflet made by the White Star Line with pictures of both being built, which stated that "as far as it is possible to do so, these two wonderful vessels are designed to be unsinkable". There's also a much nicer brochure from 1911, when the ships were closer to being finished, which touted the safety features:

Each door is held in the open position by a suitable friction clutch, which can be instantly released by means of a powerful electric magnet controlled from the Captain’s bridge, so that, in the event of an accident, or at any time when it may be considered advisable, the Captain can, by simply moving an electric switch, instantly close the doors throughout, practically making the vessel unsinkable.

The last place it was described as "unsinkable" is an issue of the trade magazine, The Shipbuilder, in 1911. This article copied the promotional text above almost word-for-word, changing the end to "make the vessel practically unsinkable".

The thing is, none of these texts were widely read. The pamphlets printed by the White Star Line were only two out of many; the first one only has one copy existing now, so it was probably in very limited circulation, too. The Shipbuilder was expensive, and only of interest to engineers rather than the general public. And all three hedge this claim of unsinkability, using it in slight hyperbole and as a statement of intent rather than a serious claim that Titanic and Olympic could never sink.

Where the "unsinkable" claim entered the mythos is in the days after the sinking. What seems most likely to have happened is that newspaper offices had kept pieces of promotional material filed for use in researching future stories, and a few journalists pulled out clippings that included one or both of these pamphlets. Philip Gibbs's "The Deathless Story of the Titanic" was published soon after the sinking, and attributes the idea of the watertight doors and compartments "practically making the vessel unsinkable" (note the similarity in the phrasing) to an official description, as did the Daily Graphic's special memorial issue, published five days after the event. Another cause is that the VP of the White Star Line, Philip Franklin, stated the morning after the sinking - when the details were still not fully known and rumors were flying about, some even claiming that the ship had been towed to Halifax - "We place absolute confidence in the Titanic. We believe that the boat is unsinkable." Where the earlier material might have been stuffed in the back of a drawer or a filing cabinet, this was said to the public, and very quickly it was multiplied into a widespread belief by the Line's leaders, the passengers, the engineers, and the world that the ship could not possibly sink because of its state-of-the-art technology. A narrative of overweening pride that was smashed by the deaths of more than a thousand innocents appeared almost overnight, helping to make meaning out of the enormous tragedy and make it feel deserved.

For further reading, I would suggest The Myth of the Titanic, by Richard Howells (1999).

u/mimicofmodes gave a really great answer, and I'd like to just add some additional information if they don't mind :)

A small correction to start- Titanic and Olympic were touted as unsinkable as early as the fall of 1910. We have a surviving official White Star Line brochure that reads- "these two wonderful vessels are designed to be unsinkable". Except, I left out the beginning of that sentence which is, "and as far as it is possible to do so". I think you can already see where this is going :) This was also reprinted in the New York Times that October in an article on the construction of Olympic-

In short, so complete will be the system of safeguarding devices on board this latest of ocean giants that, when she is finally ready for service, it is claimed that she will be practically unsinkable and absolutely unburnable.

it is claimed

To the "Shipbuilder" articles referenced- it's important to note that these don't seem to be original, and while "Shipbuilder" was an independent magazine, those articles are pretty obviously pulled from the brochures supplied by WSL itself.

So, did White Star advertise Titanic and Olympic as unsinkable? Technically, no. They wrote an advertising pamphlet that stated for all intents and purposes they were, which was reprinted as that they "practically" were, and here's all the reasons "why" they were but no- we aren't actually saying they are unsinkable :)

It is correct that "practically unsinkable" became "unsinkable" immediately, which I think is a testament to how strongly that particular phrase through strength of advertising had entered culture in the years leading up to the Titanic event. Shipbuilding was big news, really big news, and even before the two giants began to take shape- news of their supposed amazing features began to be published, including reports of being over 1000 feet long, having a golf course, and being.... practically unsinkable.

And we have good evidence of the strength of this advertising campaign. Carpathia second officer James Bissett recalls a tour of Olympic only days before Titanic set sail where an officer described her as "she was unsinkable". (Officer source is vague but I could make a stab if you wanted).

The first reports of the sinking ran with this, as we see in the Washington Times, April 16, as Titanic's fate was still not fully known-

That Captain Smith believed the Titanic and the Olympic to be absolutely unsinkable is recalled by a man who had a conversation with the veteran commander on a recent voyage of the Olympic.

The talk was concerning the accident in which the British warship Hawke rammed the Olympic.

"The commander of the Hawke was entirely to blame," commented a young officer who was in the group. "He was 'showing off' his warship before a throng of passengers and made a miscalculation."

Captain Smith smiled enigmatically at the theory advanced by his subordinate, but made no comment as to this view of the mishap.

"Anyhow," declared Captain Smith, "the Olympic is unsinkable, and the Titanic will be the same when she is put in commission."Why," he continued, "either of these vessels could be cut in halves and each half would remain afloat indefinitely. The non-sinkable vessel has been reached in these two wonderful craft."

"I venture to add," concluded Captain Smith, "that even if the engines and boilers of these vessels were to fall through their bottoms the vessels would remain afloat."

A source that wouldn't even pass muster in a high school essay, "an unnamed man who spoke to him once awhile ago" and "a young officer", and yet that was published in the immediate aftermath, cementing both the phrase and the subsequent irony and symbolism Titanic would be known for. Consider the absurdity of Smith's supposed statements- that was reported as top news in one of the country's biggest papers.

So, it seems a little bit of the hubris that Titanic became famous for, while mostly not really true, does actually have a root here. It is absurd, of course, to believe that any ship is "unsinkable" but it's not absurd to say it is with a few caveats in front :)

After Titanic, Olympic faced mutiny and went through a series of serious re-designs that countered all the factors that resulted in the loss of her sister. After all this, she was sent out again, with her new safety features being blasted across the press.

I'll give you one guess what word they used to describe her :)

EDIT: I wanted to add some food for thought, which may seem absurd but (at least I think) may have some merit. All the publicity touting the Olympic Class Liners as "practically unsinkable" was right- they were. Even though 2/3 of them sank.

Very important to remember how long Titanic took to sink- almost 3 hours. If we look at ship sinkings roughly circa 1912, we see that all of them went down in minutes, many less than 10. Her closest comparison- the Empress of Ireland capsized and sank in 14 minutes two years later. Titanic stayed afloat for 3 hours, on a relatively even keel, and sank slowly- so slowly and calmly that her sinking was actually pretty boring until the last few minutes when she collapsed all at once. And this sinking only happened because the damage she so took was eerily, precisely, lethal. Titanic's damage wasn't massive by any means, it was just surgical in how deadly it was. She very well could have survived any other collision.

Which I support by pointing to her sister Olympic, who ran into ... well.... everything really. Olympic suffered incredible damage during her years at sea, multiple collisions, survived a war, and was scrapped in the 30's with the name "Old Reliable". The damage she took, multiple times actually, dwarfed Titanic's relatively small tears and she stayed afloat- and kept sailing.

Britannic, the other doomed sister, sank after hitting a mine off the coast of Greece in 1916. To compare, Lusitania took a torpedo in roughly the same area and lasted 20 minutes. Britannic took an hour and didn't capsize.

Now, yes, I am aware that is a massively imperfect comparison that ignores the millions of subtleties between the two events. It's impossible of course to have an exact comparison, but I think it's worth noting how long the White Star ships took to sink, which I think is a testament to their incredible design (as odd as it sounds). No ship is unsinkable, of course, but in 1912- the OLC were as close as we'd ever come to making one.

That, of course, is my opinion so take it or leave it- but something to consider maybe :)