  In this work, we obtain exact thick brane models in $4+1$ dimensions
generated by higher order field theory kinks, inspired by specific potentials
for $\phi^{10}$ and $\phi^{18}$ models. We verify that the geodesic equation
along the fifth dimension confirms the confining effects of the scalar field on
the brane for all of these models. These models provide new solutions with
exponential and power-law tails which live in different topological sectors. We
show that the resulting branes of specific exponential law models do not
possess $Z_2$-symmetry. Furthermore, we examine the stability of the thick
branes, by determining the sign of the $w^2$ term in the expansion of the
potential for the resulting Schr\"{o}dinger-like equation. It turns out that
two of the three models of the $\phi^{10}$ brane are stable, while another
contains unstable modes for certain ranges of the model parameters. We also
show that the brane solution from the specific $\phi^{18}$ models are stable,
while the others involve neutral equilibrium. The asymptotic behaviour of the
brane solutions are also discussed.
