= Solution
As usual, the bounded set is assumed Lebesgue measurable and an $L^q$ exponent is positive; the conventional Banach-space assertion concerns $1\leq q<2^*=2n/(n-2)$. No boundary regularity of $\Omega$ is needed. Choose a smooth compactly supported cutoff $\eta$ equal to one on a ball containing $\Omega$. <Weak convergence> and the <Uniform boundedness principle> give a common $H^1$ bound for $f_j$, hence also for $g_j=\eta f_j$. These functions have a common compact support.
We prove their local $L^2$ compactness directly. Translation and the fundamental theorem along line segments give, by density from smooth functions,
$$
\|g_j(\cdot-h)-g_j\|_2\leq |h|\|\nabla g_j\|_2.
$$
For a compactly supported <mollifier> $\rho_\varepsilon$, integration of this estimate gives
$$
\|g_j*\rho_\varepsilon-g_j\|_2\leq C\varepsilon\|\nabla g_j\|_2\leq C'\varepsilon.
$$
At fixed $\varepsilon$, <Hölder's inequality> bounds $\|g_j*\rho_\varepsilon\|_\infty$ and $\|\nabla(g_j*\rho_\varepsilon)\|_\infty$ uniformly by the $L^2$ <norm> of the corresponding smoothing kernels. Their supports lie in one fixed compact set, so the <Arzelà-Ascoli theorem> makes this smoothed family precompact in $L^2$. The uniform $O(\varepsilon)$ approximation makes the original family totally bounded in $L^2$: choose a finite net for the smoothed family and add the approximation error. Completeness gives relative compactness.
Every strongly convergent subsequence of $g_j$ has limit $\eta f$, since multiplication by $\eta$ preserves weak $L^2$ convergence. Consequently the whole sequence converges strongly: otherwise a subsequence with distance at least a fixed positive number would have a further convergent subsequence and contradict that unique limit. In particular $\|\chi_\Omega(f_j-f)\|_2\to0$.
The critical <Sobolev inequality> supplies a uniform bound in $L^{2^*}$. For clarity, this estimate follows from part 2(a), not from compactness at the critical exponent. For a compactly supported smooth $v$, integrating by parts against the vector field $(x-y)/|x-y|^n$ gives
$$
|v(x)|\leq C_n\int |x-y|^{1-n}|\nabla v(y)|\,dy.
$$
Its distributional divergence is the sphere-area multiple of a point mass; applying the <Hardy–Littlewood–Sobolev inequality> with $\alpha=1,p=2$ yields $\|v\|_{2^*}\leq C\|\nabla v\|_2$. Density extends this to $H^1(\mathbb R^n)$.
For $2<q<2^*$ choose $0<\theta<1$ with $1/q=\theta/2+(1-\theta)/2^*$. <Hölder's inequality> gives
$$
\|\chi_\Omega(f_j-f)\|_q\leq\|\chi_\Omega(f_j-f)\|_2^\theta\|\chi_\Omega(f_j-f)\|_{2^*}^{1-\theta}\longrightarrow0.
$$
For $1\leq q<2$, use $\|v\|_{L^q(\Omega)}\leq|\Omega|^{1/q-1/2}\|v\|_{L^2(\Omega)}$. The same integral estimate also works for $0<q<1$ if $L^q$ is understood with its usual quasi-<norm>. Thus \b[$\chi_\Omega f_j\to\chi_\Omega f$ strongly for every positive $q<2^*$]. We have never asserted that multiplying by the possibly nonsmooth indicator preserves $H^1$.
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