Back in 2020, a group at OpenAI ran a conceptually simple test to quantify how much AI progress was attributable to algorithmic improvements. They took ImageNet models which were state-of-the-art at various times between 2012 and 2020, and checked how much compute was needed to train each to the level of AlexNet (the state-of-the-art from 2012). Main finding: over ~7 years, the compute required fell by ~44x. In other words, algorithmic progress yielded a compute-equivalent doubling time of ~16 months (though error bars are large in both directions).
On the compute side of things, in 2018 a group at OpenAI estimated that the compute spent on the largest training runs was growing exponentially with a doubling rate of ~3.4 months, between 2012 and 2018.
So at the time, the rate of improvement from compute scaling was much faster than the rate of improvement from algorithmic progress. (Though algorithmic improvement was still faster than Moore's Law; the compute increases were mostly driven by spending more money.)
As is tradition, about 5 minutes after the OpenAI group hit publish on their post estimating a training compute doubling rate of ~3-4 months, that trend completely fell apart. At the time, the largest training run was AlphaGoZero, at about a mole of flops in 2017. Six years later, Metaculus currently estimates that GPT-4 took ~10-20 moles of flops. AlphaGoZero and its brethren were high outliers for the time, and the largest models today are only ~one order of magnitude bigger.
A more recent paper with data through late 2022 separates out the trend of the largest models, and estimates their compute doubling time to be ~10 months. (They also helpfully separate the relative importance of data growth - though they estimate that the contribution of data was relatively small compared to compute growth and algorithmic improvement.)
On the algorithmic side of things, a more recent estimate with more recent data (paper from late 2022) and fancier analysis estimates that algorithmic progress yielded a compute-equivalent doubling time of ~9 months (again with large error bars in both directions).
... and that was just in vision nets. I haven't seen careful analysis of LLMs (probably because they're newer, so harder to fit a trend), but eyeballing it... Chinchilla by itself must have been a factor-of-4 compute-equivalent improvement at least. And then there's been chain-of-thought and all the other progress in prompting and fine-tuning over the past couple years. That's all algorithmic progress, strategically speaking: it's getting better results by using the same amount of compute differently.
Qualitatively: compare the progress in prompt engineering and Chinchilla and whatnot over the past ~year to the leisurely ~2x increase in large model size predicted by recent trends. It looks to me like algorithmic progress is now considerably faster than scaling.