= Solution
Let $h=n^{-1/3}$. Away from the boundary, the estimator is
$$
\widehat f_n(x)=\frac1{nh}\sum_{i=1}^n
\mathbf1_{\{|X_i-x|\leq h/2\}}.
$$
Its variance is at most
$$
\frac1{nh^2}\mathbb P(|X_1-x|\leq h/2)
\leq\frac1{nh}=n^{-2/3},
$$
and the <bias of a kernel density estimator> is bounded by $h\|f'\|_\infty/2$, so its squared bias is at most $h^2/4$. This gives the claimed bound, with room to spare, whenever the full window lies in the support.
At $x=0$, however, the assertion is false as printed. For example
$$
f(x)=\frac12e^{-x/2},\qquad x\geq0,
$$
satisfies $\sup(|f|+|f'|)=3/4<1$, but
$$
\mathbb E\widehat f_n(0)longrightarrow\frac{f(0)}2=\frac14,
$$
whereas $f(0)=1/2$. The mean-square error therefore tends to $1/16$, contradicting a bound tending to zero. The intended statement needs $x\geq h/2$, a boundary correction, or a condition such as $f(0)=0$.
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