Not playing on deity is an excellent way to not suck.
To answer your question, the AI on Deity has production advantages that make a second city by 3800 BC no problem.
Four-turn science often requires at least 20 cities, many good-sized, equipped with all of the obvious infrastructure: libraries, universities, marketplaces, banks and roads. Other than building science wonders, there's not much else you can do, once you have the income to set the science rate at 100%, and are in either republic or democracy. (It can be accomplished with a lower rate at times, by the way; and sometimes 100% isn't enough. If it's not, my next suggestion would be to add more beaker-producing cities.) I assume you know what to do to be able to afford a 100% science rate.
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