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Such a bound is impossible in general. E.g., suppose that $\Omega=[0,1]$, $g=0$, and $f=a^{-1/p}1_{[0,a]}$, where $p>0$ and $a\downarrow0$. Then for any $q\in(0,p)$ we have $\int|f-g|^q=a^{1-q/p}\to0$, whereas $\int|f-g|^p=\int|(f^p-g^p)=1\not\to0$.