Might carbon dating wrong
See below for details about the 45,000 annual varves in Lake Suigetsu. For example, polar bears that eat seals aren't getting their carbon from an atmospheric source.
The second kind are datings on contaminated samples, or on samples which are a mixture.
When this was first done, it turned out that carbon dating had been giving too-young dates for early civilizations.
Pollen found in the Greenland icecap has been carbon dated, and also dated by counting ice layers. Trees grow a thick ring in a good year, and grow a thin ring in a bad year.
For example, Stonehenge suddenly became older than the Pyramids, instead of younger.
Since then, several other calibrations have been done, which confirm and extend the tree-ring one.
In short, unless you have evidence to the contrary, you should assume that most of the carbon in a fossil is from contamination, and is not originally part of the fossil. The nuclear tests of the 1950's created a lot of C14.
Also, humans are now burning large amounts of "fossil fuel".
It is produced in the upper atmosphere by radiation from the sun.