If you're going for dominaion, the caaumulation of wonders and great works and the weakening of other nations culture makes it so that you can trigger a culture victory "by accident", expecially if you take out the strongest civs first.
I also had a diplomatic win before I could get enough tourism for my first culture win. I had consulates, sent trade routes to those that requested it, and got "xxx is in awe of you", which meant I had enough city states for the vote. 3 turns before the last civ would be influential. I could have spread the votes out more, but honestly, who would throw a vote for world leader?