Here "is an easy counterexample to show this can be rather poorly behaved even for smooth functions". The idea is to make $f$ almost periodic with a quasi-period $T$ (but with a slowly decreasing amplitude near $\pm\infty$, to satisfy the condition $f\in L^p(\mathbb R)$), and then shift $f$ by $T$. Then the shifted function $f_T$ will differ little from $f_0=f$.

For instance, let 
$$f(x):=c(1+\sin x)e^{-c^2 (x-\pi)^2},$$
where $c>0$ is small. Then $\|f\|_1=\sqrt\pi$ for all real $c>0$. However, 
$$\frac{\|f_{2\pi}-f_0\|_1}{2\pi}=\frac{\text{erf}(c\pi)}{\sqrt\pi}\to0$$
as $c\downarrow0$. So, there is no nontrivial (that is, nonzero) lower bound here. 

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Here is the graph $\{(x,f(x))\colon|x|<11\pi\}$ for $c=1/100$: 

[![enter image description here][1]][1]


  [1]: https://i.sstatic.net/2e6Le.png