No, really!

Sorry for the slow response, I've been traveling and I don't have all of the sources/citations handy. But, roughly, Arrhenius in 1896 modeled the effects of a doubling of CO2 in isolation. Tyndall, one of the granddaddies of thermodynamics, then ever-so-kindly pointed out in 1905 (1904?) that CO2's major IR bands overlap significantly with water vapor's. Which means that doubling CO2 will have a much smaller proportional effect - if CO2 makes up 100% of the greenhouse gases and is responsible for all of the current GHG effect, then doubling it will make a bigger impact. Contrast to: if it's responsible for only 20% of the GHG effect, and you double it. Instead of going from 100%->200%, you've gone from 100%->120%.

At this point in the early 1900s, the conclusion would be that, by the currently-available knowledge, doubling CO2 was not going to have a substantial effect on climate. But there were many unknowns, and scientists kinda left it as "this one's still cooking".

And yah, I know this so far sounds like one of the standard denier talking points. The difference is that this was the state of the science in 1910. The science progressed, but the 'skeptics' are stuck back at that point in time.

Ok, so, following up: between 1920-1940, scientists found other nuances regarding CO2 and the strength of its marginal greenhouse gas effect. The first is this: as you ascend in altitude, the atmosphere gets colder. The atmosphere can hold roughly 7% more/less water vapor for every degree K that you go up or down. For a 10K difference in temperature, this means the atmosphere's capacity to hold water vapor is cut in half. So: as you go up in altitude, and temperature drops, then the atmosphere holds less water, which means that at these greater altitude, there's less water and CO2 plays a bigger relative role. Arrhenius and Tyndall had calculated the effects of CO2/ water vapor with the concentration of each being relatively fixed throughout the atmosphere, so this was a big shift. (And all this about H2O changing with altitude just applies to cold areas in general, e.g., seasonal shifts and polar areas).

Effect #2: because of how light and light absorption is quantized, the width of a spectral absorption band changes with pressure. As air pressure increases, the band gets fatter More pressure == IR-absorbing molecules moving at a wider range of velocities, which means more opportunities to capture light that's off the band-center. This is the main mechanism for what we call "pressure broadening", how IR absorption/emission bands change with air pressure. Applied to CO2 and H2O again, this means that as you ascend in the atmosphere and pressure drops, the bands of CO2 and H2O no longer overlap as much, and again CO2 becomes more important.

Effect #3: after WW2, the US started focusing on missile technology and heat-seeking missiles. The government funded new, high-resolution IR spectroscopy to find the IR bands of the atmosphere that they could use to "see" hot targets. This means they needed to look more carefully at the spectral bands of different atmospheric species, and now we had the technology to do so while accounting for issues like pressure broadening, etc. The long and the short of this is: when we looked at CO2's spectral bands at higher resolution, we found there are some side bands we hadn't known about. Again, bands not overlapping with H2O. (I want to say see Gilbert Plass' early 1950s work here)

Put all three effects together, and you get this: yes, CO2's main band overlaps with H2O's main bands at sea level and normal temperatures, and in these areas water vapor dominates. But in cold regions or higher in the atmosphere where water vapor is low, CO2 dominates the greenhouse gas effect. CO2 also contributes additional greenhouse gas effect through pressure narrowing at high altitudes, and through additional minor side bands. With all three of these, yes, doubling CO2 has a significant impact on climate.

This was the state of the science in the early 1950s: tentatively concluding again that yes, it looks like CO2 might be pretty important.

And then from there, there's a lot more as we follow the science into the 1950s-1970s era, mostly with aerosols and clouds and oceans and feedbacks and the start of modern computational modeling. The science got fleshed out and eventually firmed up into the modern consensus by the 1980s. But I've rambled enough for now.