I want to know if the following is possible.
Fix some measure space $(X,\mu)$. Take any $f\in L^1$. Recall that $L^1 \cap L^2$ is dense in $L^1$, i.e. for every $\varepsilon > 0$ there is some $g\in L^1 \cap L^2 $ such that $\Vert f - g \Vert_1 < \varepsilon$. If $\mu(X)<\infty$ we can bound the $L^1$ norm of $g$ in terms of the $L^2 $ norm, like so: $\Vert g \Vert_1 \leq \mu(X)^{1/2} \Vert g \Vert_2$; this is an immediate consequence of Hölder's inequality.
I want to know when you can get a similar bound in the other direction, that is, that there exists some $C$ such that $\Vert g \Vert_2 \leq C \Vert g \Vert_1$. More specifically, (under what circumstances) can we show that there exists some constant $C$ (possibly depending on $\Vert f \Vert_1 $ and $\varepsilon$) such that there exists some some $g\in L^1 \cap L^2 $ such that $\Vert f - g \Vert_1 < \varepsilon$ and $\Vert g \Vert_2 \leq C \Vert g \Vert_1$? (In other words, can we show a reversal of the usual interpolation inequality on a dense subset?)
A "reverse Hölder inequality" exists in the literature. It is studied in the theory of "weights". Essentially a weight $w$ is a positive function for which the measure $w(x) dx$ have the same boudedness properties with respect to some classes of singular integral operators. Look into [D: Theorem 7.4].
The only difference between the condition that you demand and the $A_p$ conditions imposed in weighted theory is that they tend to be "open". I.e. you can have a reverse Hölder inequality for your exponent plus some (potentially small) $\epsilon >0$.
[D]: Duoandikoetxea, Javier, Fourier analysis. Transl. from the Spanish and revised by David Cruz-Uribe, Graduate Studies in Mathematics. 29. Providence, RI: American Mathematical Society (AMS). xviii, 222 p. (2001). ZBL0969.42001.