I just won my first culture victory yesterday in 450 turns on prince level, after failing once! I am posting this as I really doubted it could be done with 3 cities only.
I played with France being suitable for culture victory as it possesses a special boost toward culture points gaining (+2 culture per city). Unfortunately it vanishes after discovering steam power, so avoid discovering it as long as possible.
I was lucky to find marble during first turns and founded Paris on the same tile, so I didn't have to discover mining in order to connect marble. BTW marble speeds up wonder building. To accelerate it even more, my first social policy was the one enabling similar help - buildig wonder faster. I rushed toward previously recommended culture boosting wonders. I had them all. In addition I built many other wonders.
I had only three cities: Paris, Orleans and Lyon, the other were puppets. One was specialising in production, one in science and one in money.
I went to war against neighbouring Germany and India in medieval times in order to disallow scientific escape from me resulting in capturing 1 city from each. They lacked iron, so I had no problems, at all, despite numerical disadvantage. They challenged me in coordinate action again towards the end of renesaince, but I repelled them and gained 5 additional puppets. They fought with swords aginst rifles and cannons, so winning was easy. After that I just pillaged their countriside to slow them even more and captured and sold their workers them to gain gold - until peace traty.
After my first war I concentrated in city states. I managed to keep 2 alied culture ones, 3 allied maritime and one friendly maritime. This gained a lot of food so my workers could specialise, what lead to more gold, tech and more great persons. I earned many great persons and I mainly used them for golden ages. The wonder prolonging golden ages was very useful.
During the game I was not lagging behind in technology. I built at least one wonder in each city (culture boost social policy) and almost all culture providing buildings. The only problem I had was money. I don't really know why, but in each game I play I don't have enough. Golden ages were saving the situation. I spent my money in bribing city states and upgrading units during wars.
I also did a lot of luxuries trading to increase happines. At the beginnig my hasppines was low, but it went sky high (to cca 30) after some social policies and luxuries. I didn't have and research agreemnet because I needed money and I feared to get steam power too early. At that beginning I got new social policy in cca 20 turns, at the end in 6.
So, culture victory could be done. You should not concentrate in culture only, but also in military expansion towards other civilizations and in diplomatic spree towards city states.