Well 40K culture is at what 300turn + ? I don't have a replay but I can give you a quick summary.
I met Poland, stole a worker then proceeded to harass him with an additional archer, grabbed a settler and 2 more workers out of the salt next to his capital. I then made peace but kept blocking his settler for my 4th city spot.
I opened with 4 city NC tradition. 4th city was made during NC and I bought a library in it. I was able to get ToA but that was lucky for immortal (around T60). NC was finished around T82 and then I was able to get Petra, was risky though to make it after NC. Didn't get Desert Folklore so grabbed faith from gems.
I then grabbed construction and proceeded to Education (turn 111 I believe). Attacked Poland again and grabbed Warsaw. I also made a 5th city in the south because Jungle and Brazil.
I sadly get an additional pre renaissance policy 2 turns before education (terrible timing) and open aesthetics. I should have pushed culture stronger or weaker. This bad timing made it so I was still 8 turns away from a new policy when reaching renaissance through astronomy at turn 126 urgh. I've had this happen a couple of time lately when I only get a culture CS at friend state. I really should avoid it. Late education so I do not risk the Oracle at Renaissance gambit and use the free policy for one more in aesthetics.
Happiness on 6 city continent is hard and most of my cash goes into allying Zanzibar. I get some pagodas up. I also made the HS and enhanced with divine inspiration making building Pagoda a breeze. My expansion were able to grab HG and MoH wasn't really thinking I'd get those because I made them late but I didn't have much else to build while waiting for Education.
I spawned a GE amusingly due to ToA + Petra. Rushed Notre Dame with it. Then the rest is only about getting more wonders and trying to build Operas everywhere for the Hermitage. I reach schools arount T165 and Oxford Radio for a T170 ideology. Get freedom then I go for Hotels.
I win the world fair, get more into freedom and aesthetics beginning a golden age that I will boost with a GA at some point (missed Taj Mahal...) to make it last until the end game. I get a few themes wonder, build hotels make archeologists and reach arount 220 tourism. I see India needs only 6K more tourism (for a total of 10K) so I faith buy 2 GM, get a natural one and bomb the 3 on him. Other civs get also influenced with the 20% effect from GM. At that point I'm also close to Internet anyway. I guess having only India as a major culture civ was a bit lucky. If there were 2 civs, I'd need internet which was not far away (I'd get porcelain, a possible faith GS and finish rationalism in 2 turns).
Culture victories have this very special problem where if you're slow it becomes harder and harder to close the gap.
-The AI will increase its stockpile of culture
-The AI will reach techs allowing it to get more culture: radio, archeologists, more work of arts.
-The AI will start to get antiquity sites, reducing yours increasing their culture
-Some WC resolutions might start to be passed that help the AI if you are not close to winning: culture for wonders for example
-You must also make sure no AI wins World Fair.
-Also try to not let them AI cultural city states.
In my experience 10K culture is fairly common on Immortal in the early 200.