  We study the existence of weighted extremal K\"ahler metrics in the sense of
Apostolov-Calderbank-Gauduchon-Legendre and Lahdili on the total space of an
admissible projective bundle over a Hodge K\"ahler manifold of constant scalar
curvature. Admissible projective bundles have been defined by
Apostolov-Calderbank-Gauduchon-T{\o}nnesen-Friedman, and they include the
projective line bundles (Hwang-Singer) and their blow-downs (Koiso-Sakane),
thus providing a most general setting for extending the existence theory for
extremal K\"ahler metrics pioneered by a seminal construction of Calabi. We
obtain a general existence result for weighted extremal metrics on admissible
manifolds, which yields many new examples of conformally K\"ahler,
Einstein-Maxwell metrics of complex dimension $m>2$, thus extending the recent
constructions of LeBrun and Koca-T{\o}nnesen-Friedman to higher dimensions. For
each admissible K\"ahler class on an admissible projective bundle, we associate
an explicit function of one variable and show that if it is positive on the
interval $(-1,1)$, then there exists a weighted extremal K\"ahler metric in the
given class, whereas if it is strictly negative somewhere in $(-1,1)$, there is
no K\"ahler metrics of constant weighted scalar curvature in that class. We
also relate the positivity of the function to a notion of weighted K-stability,
thus establishing a Yau-Tian-Donaldson type correspondence for the existence of
K\"ahler metrics of constant weighted scalar curvature in the rational
admissible K\"ahler classes on an admissible projective bundle. Weighted
extremal orthotoric metrics are examined in an appendix.
