Suppose we have two simultaneous events, $A$ and $B$, separated by a distance $L$ (the simultaneity is in frame of reference $S$). Now suppose we have a second frame of reference $S'$ moving with speed $V$ relative to $S$.
According to the invariance of intervals, the interval between events $A$ and $B$ in frame $S'$ is also $L$, but since there is also a temporal separation between events (simultaneity isn't conserved), one finds that the spatial distance between events in $S'$ is larger than $L$. In contrast, according to the length contraction prediction of special relativity, the maximum length is in the rest frame, $S$. Why the inconsistency?