This is from a problem in Dummit&Foote. Here is my attempt:
$$\left<s,r^4\right> = \{1,s,r^4,r^8,r^{12},sr^4,sr^8,sr^{12}\}$$
We only need to check if the conjugate of a generator is an element of $\left<s,r^4\right>$. There are two cases I have considered.
For elements of the from $r^n$, note that $rsr = s$, so $$r^nsr^{-n} = sr^{-2n} = sr^{16-2n}$$ is one element in $\left<s,r^4\right>$ if $n=0,2,4,6,8,10,12,14$. Now the conjugate for $r^4$ is $$r^nr^4r^{-n} = r^4$$ so $n$ is arbitary. So the normalizer contains $1,r^2,r^4,r^6,r^8$.
For elements of the form $sr^n$, $$sr^nsr^{-n}s = r^{-2n}s = sr^{2n}$$ and $$sr^nr^4r^{-n}s=sr^4s = r^{-4}=r^{12}$$ so $n=0,2,4,6,8,10,12,14$.
We concluded that $\{1,r^2,r^4,r^6,r^8,r^{10},r^{12},r^{14},s,sr^2,sr^4,sr^6,sr^8,sr^{10},sr^{14}\}$ is then the normalizer of $\left<s,r^4\right>$. Is this right?