  We investigate the first-order phase transitions of the $q$-state Potts
models with $q = 5, 6, 7$, and $8$ on the two-dimensional square lattice, using
Monte Carlo simulations. At the very weakly first-order transition of the $q=5$
system, the standard data-collapse procedure for the order parameter, carried
out with results for a broad range of system sizes, works deceptively well and
produces non-trivial critical exponents different from the trivial values
expected for first-order transitions. However, a more systematic study reveals
significant drifts in the `pseudo-critical' exponents as a function of the
system size. For this purpose, we employ two methods of analysis: the
data-collapse procedure with narrow range of the system size, and the
Binder-cumulant crossing technique for pairs of system sizes. In both methods,
the estimates start to drift toward the trivial values as the system size used
in the analysis exceeds a certain `cross-over' length scale. This length scale
is remarkably smaller than the correlation length at the transition point for
weakly first-order transitions, e.g., less than one tenth for $q=5$, in
contrast to the naive expectation that the system size has to be comparable to
or larger than the correlation length to observe the correct behavior. The
results overall show that proper care is indispensable to diagnose the nature
of a phase transition with limited system sizes.
