The question is: For $n\geq 1,f_n:\mathbb{R}^d\rightarrow\mathbb{R}$ measurable. Provide a counter example to the following: If the sequence $\{f_n\}_n$ is bounded in $L^4(\mathbb{R}^d)$ then it is bounded in $L^3(\mathbb{R}^d)$.
Since $L^4\subseteq L^3 $ this seems quite correct to me and I cannot think of a counter example.
Otherwise with the same hypothesis, I want to prove that if the sequence $\{f_n\}_n$ is a convergent sequence in $L^3(\mathbb{R}^d)$ and bounded in $L^\infty(\mathbb{R}^d)$ then $\{f_n\}_n$ is a convergent sequence in $L^4(\mathbb{R}^d)$. It seems quite intuitive but I have hard time coming up with a rigorous proof.
$L^4\subseteq L^3$ is not necessarily true when the measure space is infinite, and it is famously false for $\Bbb R^n$ measure spaces. $L^p\subseteq L^q$ is just not true for $p\le q$ in that context.
A good, intuitive example is the function $x^{-p}$. It has a convergent integral over $\Bbb R_{\ge1}$ if and only if $p\gt 1$, similar to the series $\sum n^{-p}$
Let $d=1$ and let $f_n$ be any sequence of measurable functions $\Bbb R\to\Bbb R$ converging to $x^{-2/7}$. Such a sequence exists by the simple approximation theorem, or even just by setting $f_n:=x^{-2/7}\chi_{[1,\infty)}$ for all $n$. Can you see how $f_n$ is bounded in $L^4$ but not in $L^3$?