Bipartite Problem is Log-Space Reducible To $s$-$t$ Undirected Connectivity
Here are some ideas on how to reduce checking bipartiteness to undirected non-connectivity:


A graph is bipartite iff it has no odd cycles.
Suppose we guess that vertex $v$ participates in a cycle of length $2k+1$. Using a layered graph, we can check whether it is the case using undirected connectivity.
We can also implement a disjunction ("or") using undirected connectivity.


Here are some ideas for the other direction:


Every graph can be made bipartite by splitting every edge into a path of length $2$.
What happens when we join $s$ and $t$, after splitting the edges?