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We find a general expression for the free energy of $G(\phi)=
\phi^{2k}$ generalized 2D Yang-Mills theories in the strong
$(A>A_c)$ region at large $N$. We also show that in this region, the
density function of Young tableau of these models is a three-cut problem.
In the specific $\phi^6$ model, we show that the theory has a
third order phase transition, like $\phi^2$ (YM$_2$) and $\phi^4$
models. We have some comments for $k\geq 4$ cases. At the end,
we study the phase structure of $\phi^2+g\phi^4$ model
for $g\leq \frac{A}{4}$ region.
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