Example of logarithmic length witness being easier to verify than find
Consider the decision problem which decides whether a given binary input $x$ of length $n$ is a non-palindrome.
There is a pretty standard communication complexity proof that a single tape TM requires at least $\Omega(n^2)$ time to solve this problem.
On the other hand, we can also solve this problem using a nondeterministic algorithm with a $\log(n)$ length witness $i$: the algorithm accepts whenever the $i$th bit from the start of $x$ differs from the $i$th bit from the end of $x$. Identifying the $i$th bit from the start or end of a length $n$ bitstring can be accomplished in $O(n \log n)$ time on a single tape TM.