Let us consider the following sequence of continuous functions:
$$f_j(x)=\frac{1}{1+\color{red}jdist(x,\partial\Omega)},~~~~~~x\in \Omega.$$
We have, $$0\le f_j(x)\le 1,~~~~~and ~~~~~f_j(x)=1,~~~x\in \partial\Omega.$$ furthermore, for every $x \in\Omega $, we have $f_j(x)\to 0$ thus, by convergences dominated theorem we get $$\Vert f_j \Vert^p_{L^p(\Omega)}\to0 $$
Assume that $\Omega$ is bounded of class $C^1$ and there is a trace operator from $L^p(\Omega)~~$ to $~~L^p(\partial \Omega)$ then there exists a constant $C>0$ Such that for all $f \in L^p(\Omega)$ we have: $$\Vert Tf \Vert_{L^p(\partial \Omega)}\le C \Vert f \Vert_{L^p(\Omega)}.$$
In particular, for every $j$ we have $$Area(∂\Omega)=\Vert Tf_j \Vert^p_{L^p(\partial \Omega)}\le C^p \Vert f_j \Vert^p_{L^p(\Omega)}\to0. $$ $$Area(∂\Omega)=0$$
which is absurd since $\Omega$ is a bounded of class $C^1$.