  We established a Yau--Tian--Donaldson type correspondence, expressed in terms
of a single Delzant polytope, concerning the existence of extremal K\"ahler
metrics on a large class of toric fibrations, introduced by
Apostolov--Calderbank--Gauduchon--Tonnesen-Friedman and called semi-simple
principal toric fibrations. We use that an extremal metric on the total space
corresponds to a weighted constant scalar curvature K\"ahler metric (in the
sense of Lahdili) on the corresponding toric fiber in order to obtain an
equivalence between the existence of extremal K\"ahler metrics on the total
space and a suitable notion of weighted uniform K-stability of the
corresponding Delzant polytope. As an application, we show that the projective
plane bundle $\mathbb{P}(\mathcal{L}_0\oplus\mathcal{L}_1 \oplus
\mathcal{L}_2)$, where $\mathcal{L}_i$ are holomorphic line bundles over an
elliptic curve, admits an extremal metric in every K\"ahler class.
