Note that the lag from temperature to CO2 has no real relevance to the recent rise in CO2. The recent rise is from 280 to 412 (and rising 2-3 more every year). There was no corresponding temperature rise 500 to 1000 years ago to produce that CO2 rise.
Apparently you are getting your information and possibly charts from sources which have been making valiant attempts to write the “Medieval Warm Period” out of existence. There is ample evidence that the Warm Period that lasted from around 950 until approximately the Maunder Minimum that started around 1675 was much more than a local event.
You may have misread the chart, as the chart flows left to right:today to 450000 years ago. Temperature *IS* leading CO2