  hree limited samples of high-redshift radio sources of FRII-type are used to
constrain the dynamical model for the jets' propagation through the two-media
environment: the X-ray emitting halo with the power-law density profile
surrounding the parent galaxy and the much hotter intergalactic medium (IGM) of
a constant density. The model originally developed by Gopal-Krishna & Wiita
(1987) is modified adopting modern values of its free parameters taken from
recent X-ray measurements with the XMM-Newton and Chandra Observatories. We
find that (i) giant-sized radio sources ($\sim$1 Mpc) exist at redshifts up to
$z\sim 2$, (ii) all newly identified the largest radio sources with $1<z<2$
appeared to be quasars, (iii) all of them are younger and expanding faster than
their counterparts at lower redshifts, and (iv) the above properties are rather
due to the powerful jets than peculiar environmental conditions (e.g. voids) in
the IGM. The extreme powerful jets may testify to a dominant role of the
accretion processes onto black holes in earlier cosmological epochs.
