Prove: $\int_0^{\frac{\pi}{2}}t(\frac{\sin nt}{\sin t})^4dt<\frac{\pi^2n^2}{4}$
Assume that $n\in \mathbb{N}$.
Then the ratio of sines is polynomial in cosines: $$ \frac{\sin(n t)}{\sin(t)} = \cos((n-1) t) + \cos(t) \frac{\sin((n-1) t}{\sin(t)} = \ldots = \sum_{k=1}^n \cos((n-k) t) \cdot \cos^{k-1}(t) \leqslant n $$
Since $\sin(t)$ is increasing on the interval $\left(0,\frac{\pi}{2}\right)$, and since $\frac{\sin(n t)}{\sin(t)} < n$ for $0<t<\frac{\pi}{2n}$, we have: $$ \int_0^{\tfrac{\pi}{2}} t \cdot \left( \frac{\sin(n t)}{\sin(t)} \right)^4 \mathrm{d} t = \sum_{k=1}^n \int_{\tfrac{k-1}{n} \cdot \tfrac{\pi}{2}}^{\tfrac{k}{n} \cdot \tfrac{\pi}{2}} t \cdot \left( \frac{\sin(n t)}{\sin(t)} \right)^4 \mathrm{d} t < \\ \int_0^{\pi/(2n)} t n^4 \mathrm{d} t + \sum_{k=2}^n \int_{\tfrac{k-1}{n} \cdot \tfrac{\pi}{2}}^{\tfrac{k}{n} \cdot \tfrac{\pi}{2}} t \cdot \left( \frac{\sin(n t)}{\sin\left( \frac{k-1}{n} \cdot \tfrac{\pi}{2} \right)} \right)^4 \mathrm{d} t = \\ \frac{\pi^2 n^2}{8} + \sum_{k=2}^n \int_{\tfrac{k-1}{n} \cdot \tfrac{\pi}{2}}^{\tfrac{k}{n} \cdot \tfrac{\pi}{2}} t \cdot \left( \frac{\sin(n t)}{\sin\left( \frac{k-1}{n} \cdot \tfrac{\pi}{2} \right)} \right)^4 \mathrm{d} t $$ The remaining bounding integral is not hard to compute: $$ \int_{\tfrac{k-1}{n} \cdot \tfrac{\pi}{2}}^{\tfrac{k}{n} \cdot \tfrac{\pi}{2}} t \cdot \sin^4(n t) \mathrm{d} t \stackrel{t = \tfrac{(k-1)\pi}{2n} + \tfrac{u}{2}}{=} \int_0^{\tfrac{\pi}{2}} \frac{2u+ \pi(k-1)}{2n^2} \left( \frac{1+(-1)^k}{2} \cos^4 u + \frac{1-(-1)^k}{2} \sin^4 u \right) \mathrm{d} u = \frac{3 \pi^2 \cdot (2k-1) - 16 (-1)^k}{64 n^2} $$ The upper bound then becomes: $$ \frac{\pi^2 n^2}{8} + \sum_{k=2}^n \frac{3 \pi^2 \cdot (2k-1) - 16 (-1)^k}{64 n^2 \cdot \sin^4\left( \frac{k-1}{n} \cdot \tfrac{\pi}{2} \right)} $$ It is easy to check numerically that this bound is more crude than the one you seek to establish.
Added Notice the series expansion around $t=0$: $$ \left(\frac{\sin(n t)}{\sin(t)} \right)^4 = n^4 \left( 1 - \frac{2}{3} (n^2-1) t^2 + \mathcal{o}(t^2) \right) $$ This suggests looking for a bound in the form $\exp(-(n^2-1) t^2 \alpha)$. Suppose we fix a small enough $\alpha$, such that $$ \forall_{0 < t < \tfrac{\pi}{2}} \left(\frac{\sin(n t)}{\sin(t)} \right)^4 \leqslant \exp(-(n^2-1) t^2 \alpha) $$ Then $$ \int_0^{\pi/2} t \cdot \left(\frac{\sin(n t)}{\sin(t)} \right)^4 \mathrm{d} t < \int_0^{\pi/2} t \exp(-(n^2-1) t^2 \alpha) \mathrm{d} t = n^4 \frac{1 - \exp(-\alpha (n^2-1) \pi^2/4)}{\alpha (n^2-1)} = \frac{ \pi^2 n^2}{8} \cdot \exp\left(-\frac{\pi^2 \alpha}{8} (n^2-1)\right) \cdot n^2 \operatorname{sinch}\left(\frac{\pi^2 \alpha}{8} (n^2-1)\right) $$ where $\operatorname{sinch}(x) = \frac{\sinh(x)}{x}$ and is an increasing function of $x$, therefore: $$ \frac{ \pi^2 n^2}{8} \cdot \exp\left(-\frac{\pi^2 \alpha}{8} (n^2-1)\right) \cdot n^2 \operatorname{sinch}\left(\frac{\pi^2 \alpha}{8} (n^2-1)\right) < \frac{ \pi^2 n^2}{8} \cdot \exp\left(-\frac{\pi^2 \alpha}{8} (n^2-1)\right) \cdot n^2 \operatorname{sinch}\left(\frac{\pi^2 \alpha}{8} n^2 \right) < \frac{ \pi^2 n^2}{8} \cdot \exp\left(\frac{\pi^2 \alpha}{8}\right) $$
Hint: prove that $\dfrac{\sin n t}{\sin t}\le n$ for $0<t<\frac\pi{2n}$.
Write $$ \frac{\sin(n\,t)}{\sin t}=\frac{\sin(n\,t)}{t}\cdot\frac{t}{\sin t}. $$ From $\sin t\le t$ it follows that $$ \frac{|\sin(n\,t)|}{t}\le\min(n,t^{-1}),\quad t>0.\tag1 $$ From convexity, it follows that $$ \Bigl(\frac{t}{\sin t}\Bigr)^4\le1+\Bigl(\Bigl(\frac{\pi}{2}\Bigr)^3-\frac{2}{\pi}\Bigr)t,\quad0\le t\le\frac{\pi}{2}.\tag2 $$ From (1) and (2) we get that $$\begin{align*} \int_0^{\frac{\pi}{2}}t\Bigl(\dfrac{\sin(nt)}{\sin t}\Bigr)^4dt&\le\int_0^{\frac{\pi}{2}}t\bigl(\min(n,t^{-1})\bigr)^4\Bigl(1+\Bigl(\Bigl(\frac{\pi}{2}\Bigr)^3-\frac{2}{\pi}\Bigr)t\Bigr)dt\\ &=n^4\int_0^{\frac1n}t\,\Bigl(1+\Bigl(\Bigl(\frac{\pi}{2}\Bigr)^3-\frac{2}{\pi}\Bigr)t\Bigr)dt+\int_{\frac1n}^{\frac\pi2}t^{-3}\Bigl(1+\Bigl(\Bigl(\frac{\pi}{2}\Bigr)^3-\frac{2}{\pi}\Bigr)t\Bigr)dt\\ &=n^2+\frac{(\pi ^4-16) n}{6 \pi }-\frac{\pi ^4-8}{4 \pi ^2}\\ &<\frac{\pi^2}{4}\,n^2 \end{align*}$$ if $n>2$. The cases $n=1$ and $n=2$ can be checked by direct computation.
Note
A better estimate can be obtained using the inequality $$ \frac{\sin t}{t}\le\min\Bigl(\frac{\pi}{2},1+(1-\frac{2}{\pi})t,\frac{6}{6-t^2}\Bigr). $$
