  We present a new method of proving the Diophantine extremality of various
dynamically defined measures, vastly expanding the class of measures known to
be extremal. This generalizes and improves the celebrated theorem of Kleinbock
and Margulis [{\it Invent. Math.} {\bf 138}(3) (1999), 451--494] resolving
Sprind\v zuk's conjecture, as well as its extension by Kleinbock,
Lindenstrauss, and Weiss [On fractal measures and Diophantine approximation.
{\it Selecta Math.} {\bf 10} (2004), 479--523], hereafter abbreviated KLW. As
applications we prove the extremality of all hyperbolic measures of smooth
dynamical systems with sufficiently large Hausdorff dimension, and of the
Patterson--Sullivan measures of all nonplanar geometrically finite groups. The
key technical idea, which has led to a plethora of new applications, is a
significant weakening of KLW's sufficient conditions for extremality. In the
first of this series of papers [{\it Selecta Math.} {\bf 24}(3) (2018),
2165--2206], we introduce and develop a systematic account of two classes of
measures, which we call {\it quasi-decaying} and {\it weakly quasi-decaying}.
We prove that weak quasi-decay implies strong extremality in the matrix
approximation framework, as well as proving the ``inherited exponent of
irrationality'' version of this theorem. In this paper, the second of the
series, we establish sufficient conditions on various classes of conformal
dynamical systems for their measures to be quasi-decaying. In particular, we
prove the above-mentioned result about Patterson--Sullivan measures, and we
show that equilibrium states (including conformal measures) of nonplanar
infinite iterated function systems (including those which do not satisfy the
open set condition) and rational functions are quasi-decaying.
