<blockquote>
If $M$ is a $C^{1}$ manifold and $\omega_i$, $i=1,2$ are two continuous
volume forms with the same mass, does there exist a $C^1$ diffeomorphism sending one to the other?
</blockquote>

The answer is no. The equivalent problem is: given a positive and continuous function $g$, can we find a diffeomorphism with the Jacobian equal to $g$, $\det Df=g$? A counterexample is Theorem 1.2  in:
 
D. Burago, B. Kleiner, <A HREF="https://link.springer.com/article/10.1007%2Fs000390050056"><FONT FACE="Arial">Separated nets in Euclidean space and Jacobians of bi-Lipschitz maps</FONT></A><FONT FACE="Arial">.
*Geom. Funct. Anal.* 8 (1998), 273–282. 
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