KUt' whoever knows every vixen is a vixen in the normal way does so simply on the basis of their grasp of the thought

Williamson says that there are two ways to take this, a thin way and a thick way. According to the thin way the mere fact that the person grasps the thought provides an epistemic justification. According to the thick way it's the facts that constitute the grasp of the thought that provides the epistemic lift.
Williamson's argument against the thick way there's alot more to understanding the thought involved than just semantic or linguistic considerations. Many of these are in nature not conceptual or linguistic, such as the factors that add to your competence with an expression (rather than knowledge of its meaning). If the conceptual turner wants something to go on here, she needs epistemic justification that comes from merely linguistic or conceptual material.
The thought I had was that if these are genuinly the considerations necessary for grasping a thought, then they are conceptual considerations. The conceptual turner isn't dead in the water, it's just that the conceptual material she thought she was working with turned out to be a little different.
On the other hand, if all these considerations (previously thought not to be about thought or language) turn out to be conceptual, then any use for the distinction the conceptual turner needs is dissolved. Analytic thoughts or sentences don't really have any special epistemic status in virtue of being conceptual.