15 turns (1850 AD - 1880 AD)
(Re: 20% city defense bonus...I'm not entirely sure if it works anywhere within your borders. I was always under the impression it only worked for guys who are actually on the city square, but I could be wrong. At this stage in the game, the better buffing upgrade is probably +25% against gunpowder. All the best units at this point are gunpowder units (riflemen and grenadiers). Not only is the bonus bigger, it works for attacking as well.)
Here are a couple things I did immediately when taking my turns. Traded Medicine to Mao to get him to declare war on Victoria. If they're occupied attacking each other, they won't attack us. I changed our economic civic to Environmentalism to give us +6 health in all of our cities. Looked at Medina and changed the way the specialists were set up. You've got to watch Medina whenever it grows. The computer decided to auto employ priests there instead of artists. This poisons our great person generation pool and increases the chances of us getting a great prophet instead of a great artist. It also lessens our culture growth there a little. And dude, what happened to all our workers? Did they get taken after Victoria declared war on us? I'm going to make the two we have left build cottages in Damascus instead of farms. The more money we have there, the more culture we're going to generate. Remember, based on what your slider says, our commerce there becomes either science, culture, or gold. Cancelled the grocer in Damascus. All that does is give us more gold. We need commerce/culture and military pretty much exclusively now, so I'm going to build a barracks and start building riflemen there now. We could use our extra hammers to generate additional culture there instead of military, but the increase would be marginal (when I tried it, it was something like 40 more culture a turn). So instead, it's all about survival.
Additionally I used one of the great artists to expand the borders of Damascus in order to put extra cultural pressure on London and win the border war. Lastly, I changed the slider to generate culture instead of science. We're pretty much done with science from here on out. You can see the effects of these last two changes below.
Before using great artist

After using great artist

Notice how we've now taken her bananas, stone, and cows just by using a great artist. What's even sadder is the fact that our third city has almost completely engulfed her starting capital city culturally.
Before adjusting culture slider

After adjusting culture slider

I was able to change our culture slider from 80% science to 80% culture. Note the difference. Mecca went from generating 391 culture a turn to 814. Medina went from 381 to 435. Damascus went from 484 to 592. Why did Mecca jump up so high? When we adjust the slider, what we're saying is, we want to put 80% of the commerce generated in each city into culture. Mecca is almost 100% cottages/villages/towns/etc, which all generate commerce. Damascus would have jumped up in culture per turn much more if Victoria hadn't pillaged us. It makes sence that Medina went up very little, as we're working farms primarily there, which emphasize food over commerce.
So here's what happened with the rest of my turns. Huayna and Mao made peace with Victoria, but I bribed Huayna into attacking her again, just to keep her busy. I rushed the Hindu Mandir in Medina to get the culture pumping more. Everywhere else has been focusing on building military. I kept a handful of riflemen in our three secondary cities, but have been sending everyone else to the three cultural cities for defense. I did the math and decided to take 15 turns this go around so you could finish the game. Here's where we stand right now.

Mecca is going to reach 50000 culture naturally in a couple turns. We have three great artists twiddling their thumbs in Medina right now. You'll probably want to end up using all three in Medina. There should be another great artist on the way in Medina soon, and if you need it, you'll probably want to use it in Damascus. I'll let you figure it out, just do whatever gets us to 50000 culture in the three cities fastest. I'm not sure if it'll take 10 turns or not, but go ahead and finish us out regardless. If anyone attacks us, just send a bunch of riflemen to the target city and have them sit there. We can pretty much ignore everything that happens peripherally, as long as they don't take over the city.
So that should just about do it. Let me know if you want to get another game going.