This is actually backwards. Fusion weapons are substantially higher yield because they result in more fission, partly by preventing the fission primary from blowing itself up before it has finished.
Wikipedia: "Fast fission of the tamper and radiation case is the main contribution to the total yield and is the dominant process that produces radioactive fission product fallout."
Most of the fission energy in an H-bomb comes from the massive amounts of neutrons created by the fusion bomb initiating a fission reaction in the U-238 tamper of the secondary. The primary is used for its X-Rays, which cause the incredible pressures within the secondary by ablating the surface of the cylinder. When you look at this experiment and see it uses x-rays to ablate case containing the hydrogen, causing an implosion, the purpose of the experiment is clear.
It varies quite a bit by design, apparently the USSR’s initial design was only 15-20% fusion while US designs where closer to 50% which is still apparently the most efficient option in terms of warhead size.
However it’s possible to have higher fusion ratios at the expense of a larger device for the same yield. Most notably in the case of the Tsar Bomba’s which reduced the contribution of fission and too massively reduce the amount of fallout produced.
Wikipedia: "Fast fission of the tamper and radiation case is the main contribution to the total yield and is the dominant process that produces radioactive fission product fallout."
https://en.m.wikipedia.org/wiki/Thermonuclear_weapon