Which are stronger intermolecular forces: ionic bonds or hydrogen bonds? Do strong ionically attracted molecules have stronger intermolecular forces than hydrogen bonding forces?
Hydrogen bonds are the strongest intermolecular forces. Stronger than van der Waals' forces, namely (ordained by strength):


Permanent dipole - permanent dipole (Keesom's Force);
Permanent dipole - induced dipole (Debye force);
induced dipole - induced dipole (London's dispersion force);


Strong hydrogen bonds are created by the FONs (fluorine, Oxygen and Nitrogen), which are very electronegatives, what produces a strong attraction between one them and hydrogen (which is less electronegative). As example of this interaction we can cite the water, that's a polar molecule in a "V" shape, take a look at the figure below. 

Then, the both hydrogens yields a $\delta^{+}$ that's attracted by the Oxygen (very strong electronegative, the $\delta^{-}$ side) of the another water molecule, yielding the hydrogen bonds, take a look at the figure below.