  We establish uniform bounds for the sup-norms of modular forms of arbitrary
real weight $k$ with respect to a finite index subgroup $\Gamma$ of
$\mathrm{SL}_2(\mathbb{Z})$. We also prove corresponding bounds for the
supremum over a compact set. We achieve this by extending to a sum over an
orthonormal basis $\sum_j y^k|f_j(z)|^2$ and analysing this sum by means of a
Bergman kernel and the Fourier coefficients of Poincar\'e series. As such our
results are valid without any assumption that the forms are Hecke
eigenfunctions. Under some weak assumptions we further prove the right order of
magnitude of $\sup_{z \in \mathbb{H}} \sum_j y^k|f_j(z)|^2 $.
