Captured wonders don't give you any culture.
If you want a 20k culture win (as opposed to 100k), these are some basic things to keep in mind.
0. Temples and Cathedrals in other towns don't help you win. Don't build them. Spend the shields on something useful, and you'll have money to research with instead of paying maintenance on the religious buildings.
1. If your first town is your 20k town, you cannot use a palace prebuild. If you have trouble keeping up in tech, being able to prebuild with the palace will let you sneak in ahead of the AI to snag wonders. You should be able to keep up in tech on Regent by planning research and trading appropriately, but if this is a problem, do 20k in your second town.
2. Make sure your 20k town has fresh water. You don't want to have to invest a bunch of shields into an aquaduct, but also you don't want to get stuck at size 6 until construction, which otherwise will probably be one of the last techs of the ancient age.
3. Your 20k town also needs food and shields. It doesn't need lots of extra food, just enough to let you work all the shield tiles. If it doesn't have lots of food, you'll want to add workers from faster-growing towns to build its population up quickly. If possible, you'll want to add workers to get it from size 6 to size 12 even if it has lots of food, as you want it to grow really fast.
4. Research toward the Republic slingshot (assuming you are playing Conquests), deviating only enough to make sure your 20k town has something to build. Ceremonial Burial, Bronze Working, and Masonry are important, so if you don't start with one of these, research one before heading to the Republic. Try to find opponents to trade with to get the others. Don't trade away writing, but trade starting techs whenever you can.
5. Upon learning philosophy, building the MoM can be really helpful. This will let you drop your lux rate, which has probably been quite high, trying to keep your very large 20k town happy. This, in turn, will let you up your research rate.
6. Make sure that one of your early non-20k towns is pumping out settlers to build you an empire to support your 20k town. Make sure another is pumping out military units to keep you safe.
7. Make sure you have lots of workers. You need your 20k town to be working fully-improved tiles, you need your other towns to be improved, and you need workers to add to the 20k town. Make sure everything is roaded so you have some money to research with.
8. Build cheap culture buildings before expensive ones. Don't build improvements if you can buy them, though. Sometimes you'll want to build a particular wonder first to make sure you get it, and sometimes you'll want a wonder for reasons other than culture (Knights Templar if you have no iron or saltpeter, for example), but generally make sure you are focused on the highest culture per shield improvements and wonders.
9. Later on, Shakespeare's Theater is important to build early, because it lets you grow past size 12. Beeline Free Artistry. Then you need railroads and factories to bump up your shield output.
10. Only build wonders in your 20k town. Use the other towns to support it. Don't worry about getting all the wonders. If you get 2 or 3 wonders in the ancient age, that will be fine.