How do Young Earth Creationists account for civilisations with histories going back five thousand years or more? How do Young Earth Creationists account for the fact that some cultures have been continuously measuring the years for almost five millenia?
Lining up dates from ancient documents and inscriptions is not an easy thing to do. You don't find ancient records that say, "Our nation was established on June 12, 3462 BC", for the simple reason that our modern calendar did not exist back then. So even when an ancient source gives a date, it is in the calendar they used at the time, and historians have to try to match that up to modern calendars. Scholars debate these sort of things endlessly.

So the reality is not that ancient Egyptian or Hittite or Sumerian or whatever chronologies contradict the Bible chronology. Rather, it is that SOME INTERPRETATIONS of ancient Egyptian, etc chronologies contradict SOME INTERPRETATIONS of the Bible chronology. Put that way, it's hardly an astounding fact. Reporters trying to describe current events often have difficulty fitting together all the different accounts of an event into a single coherent narrative.

When people talk about taking Bible dates literally, they usually eventually refer to James Ussher, who used the genealogies and other information in the Bible to estimate the date of creation at about 4004 BC. Modern young-earth creation theorists generally accept that as at least a ballpark, though many would stretch it out a few thousand years. Some secular historians claim to trace historical records back to circa 6000 BC. So yes, there's a discrepancy to be resolved, but it's on the same order of magnitude. No one claims to be able to trace historical records back a million years or a billion years. The histories of cultures around the world are at least generally consistent with young-earth Biblical chronology. There's no way you could use them as evidence for an evolutionary time-scale.