Let $f_n \in L^2(0,1)$ with the property that $\sup_n || f_n ||_{L^1}<A< \infty$, i.e. $f_n$ is a sequence in $L^2$ that is uniformly bounded in the $L^1$-Norm.
Does $f_n$ then have a weak converging subsequence in $L^2$?
i.e. is there a $f \in L^2$ and $n_k$, such that $ \int gf_{n_k} \rightarrow \int gf$ for all $g \in L^2(0,1)$?
As pointed out by David Mitra, the answer is no.
In fact, a sequence which converges weakly in $L^2$ is bounded in $L^2$. Conversely, a sequence bounded in $L^2$ possesses a subsequence which converges weakly in $L^2$.
Hence, at least a subsequence of your $\{f_n\}$ has to be bounded in $L^2$, which might not be true for all sequences which are bounded in $L^1$.