> In the 70s, the scientific consensus was global warming, not cooling.

Consensus is irrelevant here, in the 1800s the consensus was that radiation travelled through the aether. This did not make it true even though it was the consensus.

> This is irrelevant for the current trend

Of course it is relevant for the current trend, it is those historical events which can be used to check climate models. This goes for both recent history (where direct measurements exist) as well as ancient history (where measurements need to be derived indirectly).

> I suppose you have data to back this claim, isn't?

Why, yes, there are many examples. There are many articles pointing out that CO₂ concentration seems to lag behind temperature changes instead of the other way around:

[1] "Temperature and atmospheric carbon dioxide are significantly correlated over the past thirty years. Changes in carbon dioxide content lag those in temperature by five months."

[2] "High-resolution records from Antarctic ice cores show that carbon dioxide concentrations increased by 80 to 100 parts per million by volume 600 ± 400 years after the warming of the last three deglaciations."

[3] "The lag was calculated for which the correlation coefficient of the CO2 record and the corresponding temperatures values reached a maximum. The simulation yields a lag of (1200 ± 700) yr."

[4] "The start of the CO2 increase thus lagged the start of the [temperature] increase by 800 ± 600 years."

[5] "Over the full 420 ka of the Vostok record, CO2 variations lag behind atmospheric temperature changes in the Southern Hemisphere by 1.3±1.0 ka"

[6] "The sequence of events during Termination III suggests that the CO2 increase lagged Antarctic deglacial warming by 800 ± 200 years and preceded the Northern Hemisphere deglaciation."

More on the 'greenhouse potential (GWP)' of CO₂ (and other IR-active gases like CH₄ and N₂O) in "Analyses of IPCC’s Warming Calculation Results" [7] et al.

As to the overestimation of the temperature increase all it takes is a look at the average temperature development versus the predicted temperatures to see the discrepancy. Some articles detailing this discrepancy:

[8] "Actual temperature measurements of 1958-2017 from three weather balloon datasets were compared to all 102 model runs. ... The mid-tropospheric warming trends in the models are caused by increasing water vapour in response to rising CO₂ levels due to flawed moist convection thermodynamic parametrization. Correcting the parametrization to match the temperature and water vapour amounts would greatly reduce the model’s climate sensitivities by reducing the water vapour feedback net of the smaller lapse rate feedback."

[9] "Why models run hot: results from an irreducibly simple climate model"

There are many more articles which pose questions around the accuracy and/or validity of the models on which the climate catastrophe hypothesis is based.

[1] http://www.nature.com/nature/journal/v343/n6260/abs/343709a0...

[2] http://www.sciencemag.org/cgi/content/abstract/283/5408/1712

[3] http://www.agu.org/pubs/crossref/2000/1999GL010960.shtml

[4] http://www.sciencemag.org/cgi/content/abstract/291/5501/112

[5] http://dx.doi.org/10.1016/S0277-3791%2800%2900167-0

[6] http://www.sciencemag.org/cgi/content/abstract/299/5613/1728

[7] http://www.jcbsc.org/admin/get_fileenv.php?id=110

[8] https://agupubs.onlinelibrary.wiley.com/doi/epdf/10.1029/201...

[9] DOI 10.1007/s11434-014-0699 (Sci. Bull. (2015) 60(1):122–135)