Is deciding whether or not a graph is bipartite in $L$?
Your construction shows that checking whether an undirected graph is bipartite is logspace reducible to PATH (deciding whether there exists a path between two vertices in an undirected graph).

Recall that if $A$ is logspace reducible to $B$ and $B\in L$, then $A\in L$. To decide $A$ in logspace, you don't actually compute the reduction (as its output can be of polynomial length), but compute a single bit of the reduction whenever it is requested. More concretely, suppose $f$ is a logspace reduction from $A$ to $B$. To decide $A$, given input $x$, execute the logspace machine for $B$ and keep a counter $i$ for the location on its input tape. Whenever $B$ wants to read a symbol from its input, compute the $i$'th bit of $f(x)$, which requires only logarithmic space (as $f$ is a logspace reduction from $A$ to $B$).

This show that you don't actually need to construct $G'$ entirely. You only need to compute a given bit of its adjacency matrix in logarithmic space.