  The observed accelerated expansion of the Universe at present epoch can be
explained by some of the $f(R)$ models without invoking the existence of dark
energy or any other such exotic component in cosmic fluid. The $f(R)$ models in
Palatini formalism is relatively less explored in recent times with respect to
their counterpart in metric formalism. We study seven $f(R)$ models in Palatini
formalism: Hu-Sawicki (two cases), Starobinsky, exponential, Tsujikawa, $f(R) =
R -\beta /R^ n$, and $f(R)= R + \alpha \ln(R) - \beta$. Following standard
statistical procedure and utilizing data sets: type Ia supernovae data, cosmic
chronometer observations, baryonic acoustic oscillations data, data from H
\textsc{ii} starburst galaxies, local measurements of the \emph{Hubble}
parameter ($H_{0}$), and distance priors of cosmic microwave background
radiation data, we obtain constraints on the model parameters. When compared
with the standard `lambda-cold dark matter model', for many data set
combinations, the support for $f(R)$ models is significant. We obtain the
relevant quantities for characterizing the accelerated expansion of the
Universe, and these quantities are consistent with those obtained in a
model-independent way by others. The curve of effective/total equation-of-state
parameter, obtained from parameter constraints, clearly shows correct phases of
the expansion history: the radiation-dominated epochs and the matter-dominated
epochs, of the past, and the current accelerated expansion epoch eventually
evolving to de-Sitter phase in the distant future. Overall, our results
advocate in favour of pursuing $f(R)$ models in Palatini formalism as a
potential alternative for explaining accelerated expansion of the Universe.
