$stCON$ with path of length $≥ n/2$ The following problem seems very similar to the $stCON$ decision problem: 

{$G, s,t  |  G = (V, E)$ such as $V$ is a graph, $s,t ∈ V$,
there exists in $G$ a simple path from $s$ to $t$ of length $≥ n/2$}

But from what I've read, it cannot be placed in $NL$. The reason is unclear. Why it cannot be placed in $NL$ using an verifier Turing Mechine like in $stCON$ case and another counter for the path length?
The NL algorithm for $s$-$t$ connectivity proceeds as follows. Start at $s$. For $n-1$ steps, guess a neighbor of the current node, and accept if that neighbor is $t$. The memory requires is two vertices (the current one and the next one), and a counter that counts the length of the current walk.

If you want to check that there is a simple $s$-$t$ path of length at least $n/2$, you need some way to ensure that in the process, vertices don't repeat. It's not so clear how to do that without storing all vertices visited so far, which cannot be done in logspace.