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Let
$$ \sigma = (1\ 2\ 4), \tau =(1 \ 2 \ 5 \ 3) \ (4\ 7\ 6) $$
where $\sigma, \tau \in S_7$. Based on e.g. this answer, I think it should be the case that:
$$ \tau \sigma \tau^{-1} = (\tau(1)\ \tau(2)\ \tau(4)) = (2 \ 5 \ 7) $$
but when I work it out, I find that:
$$ \tau \sigma \tau^{-1} = (1\ 6\ 3) $$
and:
$$ \tau^{-1} \sigma \tau = (2\ 5\ 7) $$
but:
$$ (\tau^{-1}(1)\ \tau^{-1}(2)\ \tau^{-1}(4)) = (3\ 1\ 6) $$
Where am I going wrong here?
You have to be careful about whether you are combining your permutations from left-to-right or right-to-left, which corresponds to whether you are thinking of $S_n$ acting on $\{1,\dots,n\}$ on the right or left respectively.
To illustrate if we combine the permutations from left-to-right we get $$\tau\sigma\tau^{-1}=(1253)(476)(124)(1352)(467)=(163)$$ and $$\tau^{-1}\sigma\tau=(1352)(467)(124)(1253)(476)=(257).$$
However, if we combine from right-to-left we get $$\tau\sigma\tau^{-1}=(1253)(476)(124)(1352)(467)=(257)$$ and $$\tau^{-1}\sigma\tau=(1352)(467)(124)(1253)(476)=(163).$$
Usually if you call $\tau\sigma\tau^{-1}$ "the conjugate of $\sigma$ by $\tau$" you are imagining $S_n$ acting on the left and so you get the answer $(257)$ as stated in the answer you reference. I suspect that you are tacitly using the opposite convention which explains the discrepancy.
Note that your final equation is consistent because as permutations $(163)=(316)$.
