Without opening your save, let me ask some questions about your basic culture strategy. For reference, I normally get culture wins in the mid 1600s on Settler, mid-late 1700s on medium difficulties, and early 1900s on deity duel maps.
What civs are you playing as? If you want to speed up culture, Siam, Egypt, and France are great choices, with Aztecs also a great choice if you want a war-heavy game. Map type and game speed aren't going to matter as much, though land-heavy maps will be a bit faster just because you'll have more room to grow your cities and usually meet your rivals and the CSs faster.
Early game: Get at least one scout out first, looking for ruins. Don't use the auto-explore function. Then build a monument, start on a granary. Research pottery - calendar - writing and then philosophy on easier and harder difficulties (on mid-level, I find it's better to get a few others first and then use GL for philosophy). I usually end up buying the temples.
Policies: Take the Tradition opener for the +3 culture in your capital, then Liberty. Get yourself the worker and settler from the first two Liberty policies, then go take the Aristocracy tradition from Tradition for faster wonder production. You'll mostly be building wonders in your capitol for a bit, so this will help a lot.
Do NOT take Legalism yet. You'll want to save that for Opera Houses. Get the remainder of the Liberty policies, which will give you a free GP of your choice. I almost always pick an Engineer, who rushes a wonder in my lowest-producing city. Next take all of the Piety policies. By this time you should be almost to the Renaissance, if not there yet. Acoustics should be your first Renaissance tech, which will give you access to Opera Houses. Don't build them, they're expensive. Instead, as soon as possible AFTER getting Acoustics, take Legalism, which will give you 4 Opera Houses instead of a few monuments or temples if you'd taken it earlier. You'll want to get all of the Freedom policies next, which will give you double output from great people tile improvements. The last policy branch is up to you, though Patronage is a good choice.
You're right that you don't want too many cities. You also don't want too few though. Four is ideal, since you get the four free culture buildings from Legalism, but three's usually fine too. If you really want more cities than that, acquire them as puppets, since they don't drive up policy costs.
How do you normally plan your techs? I've found that the best tech path is to follow the useful culture wonders, shooting for Philosophy first, then aim for Chivalry (Himeji Castle is +3
and provides a nice defensive bonus if you think you'll need it, but mostly gets you to Acoustics quickly. After Acoustics, take the sailing techs up through Navigation and then pick up Archaeology for the last major cultural buildings you'll need. Then go back and pick up the bottom of the tech tree as you find convenient. Make sure you get workshops and the ironworks in your capitol or other major production city to speed up the Utopia project when the time comes.
Use all Great Artists for monuments in your capitol to take advantage of the +%culture from Hermitage. Maybe save one right towards the end to make sure you'll have Golden Age turns to build Utopia.
How are you handling diplomacy? On Settler, it usually takes quite a bit of work to get someone to DoW you, so unless you really want that for some reason, you can pretty much ignore military. Higher up you'll need a few units, but still not many if you don't actively try to piss people off. I've won duel deity games where I didn't build a single military unit until the late Renaissance except early scouts and a caravel for exploration. City States are obviously key. The maritime ones can actually be as beneficial as the cultural ones, as they allow for fast middle-game growth (yielding more gold and production, and giving you food so you can have as many specialists working your temples/monasteries/etc).
The end of the game really depends on how much tech you'e gotten. You can go low-science, where after speeding to Acoustics and Archaeology you mostly ignore science. The other option is to keep pressing forward with science, aiming to get the free tech from the Sydney Opera House. There's a very direct path through the Industrial techs if you just want culture. You only need Bio, Steam, Electricity, and Radio before Mass media becomes available. If you can hard-research two of those, and save up any GS's you get during the game plus save Oxford university, you can often buzz through the Industrial Age very quickly, then build SoH or use a GE to rush it for what should be your last or second-to-last policy. Note that you'll need a few hammers, since a GE can only rush up to 1000 hammers, and SOH is around 1050. Note that Cristo Redentor is not nearly as nice as it used to be since the July patch, and the extra culture from radio towers usually does way more for you than CR would.
What wonders to you try for? Here's my mandatory/nice lists:
Mandatory:
Stonehenge
Oracle
Sistine
Really Really useful, nearly mandatory:
Hagia Sophia - get a GE, use it to rush another wonder in a low producing city (for bonus from 3rd Piety policy)
Hanging Garden
Louvre
Great Library
Notre Dame (Note that the + 10 happy is also +5
with bonus from Piety policy).
Nice if you can get them:
Pyramids (only good if you can get it in 2nd city reasonably early; generally not worth building in capitol).
Angkor Wat
Chitzen Itza
Porcelain Tower, especially if going for higher tech option mentioned above.
Great Wall (early + 3 culture)
Himeji Castle & Kremlin - mid and late-mid game +3 culture
Good luck!
