  The problem of whether and how one can compute the twin-width of a graph --
along with an accompanying contraction sequence -- lies at the forefront of the
area of algorithmic model theory. While significant effort has been aimed at
obtaining a fixed-parameter approximation for the problem when parameterized by
twin-width, here we approach the question from a different perspective and
consider whether one can obtain (near-)optimal contraction sequences under a
larger parameterization, notably the feedback edge number $k$. As our main
contributions, under this parameterization we obtain (1) a linear bikernel for
the problem of either computing a $2$-contraction sequence or determining that
none exists and (2) an approximate fixed-parameter algorithm which computes an
$\ell$-contraction sequence (for an arbitrary specified $\ell$) or determines
that the twin-width of the input graph is at least $\ell$. These algorithmic
results rely on newly obtained insights into the structure of optimal
contraction sequences, and as a byproduct of these we also slightly tighten the
bound on the twin-width of graphs with small feedback edge number.
