Why does Saturn have a more prominent equatorial bulge and higher flattening ratio than Jupiter despite rotating slower?
Consider the dimensionless parameter $k$ for some planet, given by
$$k \equiv \frac{\Omega^2 a^3}{G M}$$
where


$\Omega$ is the planet's equatorial sidereal rotation rate,
$a$ is the planet's equatorial radius,
$G$ is the universal gravitational constant, and
$M$ is the planet's mass.


The values for this dimensionless parameter are


0.003461 for the Earth,
0.08919 for Jupiter, and
0.1576 for Saturn.


The factor $k$ is essentially the ratio of centrifugal effects to gravitational effects at the "surface" at the equator of the planet, surface in scare quotes because giant planets don't quite have a surface. Ignoring higher order terms, the flattening of a uniform density planet in hydrostatic equilibrium can be shown to be $\frac54k$. The mathematics that go into this derivation are a bit hairy, even for a planet of uniform density, so for now I'll just state this as a fact. (But see Rotational Flattening, for example. Note well: The derivation at this site involves a bit of hand-waving. Overcoming the hand-waving involves even more mathematics.)

The assumption of uniform density is not that good. Even if one does allow for density variations as a function of radial distance from the center of the planet, the parameter $k$ defined above plays a significant role in the computation of what the flattening should be. The observed flattening of Saturn versus that of Jupiter is consistent with the fact that the value of $k$ for Saturn is 1.76 times that of the value for Jupiter.