Why is it that every k-tape Turing machine has a 1-tape TM that runs in $O(t^2(n))$?
In your suggestion you write the tapes next to one another. That has another disadvantage, if one of the tapes runs out of space one has to move the information to extend that space.

The solution is to write different tapes on "tracks", above each another, technically this is done by extending the alphabet to a cartesian product of alphabets. If the original tapes all have alphabet $\Sigma$ (for $i=1,\dots,k)$ and blank $B\in \Sigma$, the single tape simulator has alphabet $\Sigma^k\cup\{B\}$. Additionally one needs to mark the positions of the heads, leading to extra symbols on the tracks.

(edited to include answer to question in comment)