  In this paper, we quantify the ability of multiple measurements of
high-redshift quasars (QSOs) to constrain several theories of modified gravity,
including the Dvali-Gabadadze-Porrati braneworld scenario, generalized
Chaplygin gas, $f(T)$ modified gravity, and Modified Polytropic Cardassian
model. Recently released sample of 1598 quasars with X-ray and UV flux
measurements in the redshift range of $0.036\leq z \leq 5.1003$, as well as a
compilation of 120 intermediate-luminosity radio quasars covering the redshift
of $0.46 < z < 2.76$ are respectively used as standard probes at higher
redshifts. For all considered modified gravity theories, our results show that
there is still some possibility that the standard $\Lambda$CDM scenario might
not be the best cosmological model preferred by the current quasar
observations. In order to improve cosmological constraints, the quasar data are
also combined with the latest observations of baryon acoustic oscillations
(BAO), which strongly complement the constraints. Finally, we discuss the
support given by the data to modified gravity theories, applying different
information theoretic techniques like the Akaike Information Criterion (AIC),
Bayesian Information Criterion (BIC) and Jensen-Shannon divergence (JSD).
