  In this paper, we study quasilinear parabolic equations with the nonlinearity
structure modeled after the $p(x,t)$-Laplacian on nonsmooth domains. The main
goal is to obtain end point Calder\'on-Zygmund type estimates in the variable
exponent setting. In a recent work \cite{byun2016nonlinear}, the estimates
obtained were strictly above the natural exponent $p(x,t)$ and hence there was
a gap between the natural energy estimates and the estimates above $p(x,t)$
(see \eqref{energy} and \eqref{byunok}). Here, we bridge this gap to obtain the
end point case of the estimates obtained in \cite{byun2016nonlinear}. To this
end, we make use of the parabolic Lipschitz truncation developed in \cite{KL}
and obtain significantly improved a priori estimates below the natural exponent
with stability of the constants. An important feature of the techniques used
here is that we make use of the unified intrinsic scaling introduced in
\cite{adimurthi2018sharp}, which enables us to handle both the singular and
degenerate cases simultaneously.
