(not a climate scientist)
Also, for the graph showing "peaks" - it seems to me that the "pattern of peaks" is constant through your graph, but the average temperature rises continually through the peaks.
If I were to guess, the peaks are things like volcanoes, currents and such, which are "constant" through time. The rise in CO2 is shown in the graph by the median not being a flat line.
If you have a step function and draw a trend line through it, the trend line will go up, but that doesn't imply a continuous process. CO2 forcing is (asserted to be) a continuous process. More co2 = higher temperature. The attempts to explain why that's observably not the case have become particularly wild in recent years, yet we are constantly told the science is settled and other untrue things.
Co2 is far from the only thing affecting temperature. The graph you're posting is clearly not a step function either. More like some kind of oscillation with an additional linear trend. Maybe the oscillation is el nino, it seems to be roughly periodic to 7 years or so, I don't know.
Edit: the bigger peaks are bang on el nino years, especially the '98 peak
Yep the step function is El Nino. There is also La Nina which can cool things, the AMO and other natural inputs that affect temperature. So it turns into a debate about how much each factors matters, which is the big set of unknowns that have no clear answer. Hence why the science isn't settled.
Modelling assumes it's carbon, that's taken as a premise. The models are built to calculate that the climate is stable if not for industrial activity. If the climate is simulated as non-stable even in pre-industrial times, that's assumed to represent a bug in the model.
Technically speaking, gradual increases and decreases over time don't have to be explained for CO2-doom to be wrong. Invalidating a theory doesn't require replacing it with a different one. But it could be AMO or sunspot activity or many other things that were once considered uncontroversially to have a big impact on the climate.
National Geographic, 1967:
Dr. Hurd C. Willett, Professor of Meteorology at Massachusetts Institute of Technology, suggests the answer. Dr. Willett, one of our staff affiliates this year, has shown us how cyclic changes in the climate closely parallel the cyclic changes in sunspot activity—the manifestations of powerful electrical energy discharges from the sun. We now feel confident that our investigations here back up the solar-climate theory of weather cycles.
If I were to guess, the peaks are things like volcanoes, currents and such, which are "constant" through time. The rise in CO2 is shown in the graph by the median not being a flat line.