The "west that is forgotten" is the Undying Lands.
I imagine that over the years, the idea of a place where the Elves rest for eternity would retreat into legend since Elves became more introverted among mortal Men and the knowledge of their existence and fate was less known.
Especially to men who aren't necessarily of Numenorean descent.
scifi.stackexchange.com/users/46713/John Bell
I think you're parsing that wrong.
It's not the West that is forgotten, but Gandalf's youth there.
scifi.stackexchange.com/users/969/Daniel Roseman
Taking it as written, it is the West that is forgotten.
It's a similar construction to "blade that was broken."
I took it to mean the majority of regular folk, hobbits, non-noble dwarves, and men of Gondor and Rohan, and all those who live beyond the edges of Tolkein's map, do not know about the Undying Lands.
(Although most of the characters we become intimate with: wizards, kings, elves, and more-sophisticated hobbits, have not forgotten the West and know its importance, it "is forgotton" by most folk living at the end of the Third Age.)
To me it also hints at Gandalf's understanding that such knowledge will eventually be completely forgotten by all who remain in Middle Earth.
The decline and fall that is a strong theme of Tolkien's work.
And if you take the tales of Middle Earth at their stated (by the author) purpose, as millennia-old tales of a mythic Britain, he's talking about our days, long after all the Elves left, and dwarves and hobbits faded or merged with Men.
No doubt I'm reading too much into such a brief expression.
scifi.stackexchange.com/users/24894/Dan Barron
The only people who care about Valinor ("the West") in The Two Towers are the remnants of the Noldorin exiles in Rivendell, Loth-Lorien and the Grey Havens.
We know from Frodo's accounts that some of these elves have been traveling through the Shire on their way to the Grey Havens to catch the last ships into the West, and we know that their leaders leave soon after the events of the Lord of the Rings.
The remaining elves -- the Moriquendi -- have no first-hand knowledge of the West, and no desire to travel there (unless, like Legolas, they glimpse the sea and are overcome by sea-longing ).
Thus, except for the very few Noldorin exiles left after the last ship sails into the West, Valinor will be completely forgotten --
as indeed it would have been, if the Red Book of Westmarch had not been written cataloging these stories.
scifi.stackexchange.com/users/274/Gaurav
I always interpreted it as Gandalf's youth (in the West) that was forgotten, since all Elves and Dunedain remembered Valinor.
Any elf who had not definitely refused the call to go to Valinor - planning to say in Middle-earth and fade away instead - intended to go to Valinor someday in the near or very distant future.
The elves had been told time and time again that it was their destiny to go to Valinor or else fade, and they could see the world becoming less and less hospitable to them over the millennia.
Galadriel was the only exile who was still forbidden to return to Valinor at the time of LOTR, and she certainly had not forgotten Valinor.
She sang about wondering if a ship could ever now come to carry her back across so wide a sea.
And the Dunedain could not forget about Valinor without forgetting the reason for the downfall of Numenor, the most important part of their origin myths and legends.
In Gondor they still stood in silence and gazed at the west before meals, looking toward Numenor that was, Elvenhome that is, and Valinor that ever shall be.
The west was not forgotten but was a vital part of the intellectual and spiritual lives of millions of elves and Dunedain and non Dunedain Gordorians -
the population of Gondor should have been a few million.
But certainly even Elven and Dunedain lore masters would seldom have mentioned or thought about the deeds of Olorin in Valinor, since he mostly worked unseen by elves.
Gandalf's (relative) youth as Olorin in the West was largely forgotten and at most barely a footnote in Middle-earth.
Added 08-14-2019.
See also here:
When did Bilbo and Frodo learn that Gandalf was a Maia? 1
And here: Who in Middle-earth knows the Istari's origin?
2
And here: Why does Gandalf say "the West that is forgotten"?
3
Why does Gandalf say "the West that is forgotten"?
3
scifi.stackexchange.com/users/43538/M. A. Golding
There are also remarks in "Unfinished Tales" and "The Silmarillion" concerning the Istari (Wizards) and specifically, but briefly, Olorin (Gandalf).
In the Undying Lands he was of the race of beings that were worshiped by the Elves.
I had always assumed that as Gandalf in Middle-earth
he was "clothed" in a human form, and that that necessitated a lessening of his power and knowledge.
Note his curious lack of memory when he first returns from death and meets with Aragorn, Legolas, and Gimli in the eaves of Fangorn.
scifi.stackexchange.com/users/100092/Chronicnecrosis
Inspired by this  question  and namely by this passage (emphasis mine):   “Many are my names in many countries: Mithrandir among the Elves, Tharkûn to the Dwarves; Olórin I was in my youth in the  West that is forgotten , in the South Incánus, in the North Gandalf; to the East I go not.”   Why does Gandalf say that West is forgotten? What does he mean?
Why does Gandalf say "the West that is forgotten"?
scifi.stackexchange.com
<the-lord-of-the-rings><tolkiens-legendarium><gandalf>
scifi.stackexchange.com/questions/111887
scifi.stackexchange.com/users/40938/Sok Pomaranczowy
Gandalf speaks of "the West that is forgotten" to refer to the Undying Lands. It appears to be that other characters, mainly the elves, have no first-hand knowledge of "the West" or aren't aware of them at all. However, it is also possible that Gandalf refers to the west as "forgotten" because he himself cannot recall his memories of it. The "West that is forgotten" most likely means the lands in the West where the elves now live which have been forgotten by many of the other creatures in Middle Earth.
The west was not forgotten but was a vital part of the intellectual and spiritual lives of millions of elves and Dunedain and non Dunedain Gordorians - the population of Gondor should have been a few million.
It means the undying lands where the elves now live but which men and other creatures have forgotten. It refers not to tge forgotten West but to Gandalf's forgotten youth in the West.