  In this article, we study the Mahler measures of more than 500 families of
reciprocal polynomials defining genus 2 and genus 3 curves. We numerically find
relations between the Mahler measures of these polynomials with special values
of $L$-functions. We also numerically discover more than 100 identities between
Mahler measures involving different families of polynomials defining genus 2
and genus 3 curves. Furthermore, we study the Mahler measures of several
families of nonreciprocal polynomials defining genus 2 curves and numerically
find relations between the Mahler measures of these families and special values
of $L$-functions of elliptic curves. We also find identities between the Mahler
measures of these nonreciprocal families and tempered polynomials defining
genus 1 curves. We will explain these relations by considering the pushforward
and pullback of certain elements in $K_2$ of curves defined by these
polynomials and applying Beilinson's conjecture on $K_2$ of curves. We show
that there are two and three explicit linearly independent elements in $K_2$ of
certain families of genus 2 and genus 3 curves.
