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One of the biggest reasons is that hydrogen bonds are intermolecular forces, which happens between two atoms in the same molecule. A hydrogen bond is merely an electromagnetic attraction, not really a bond.
Since there is no actual electron exchange taking place (whether by the sharing of electrons in covalent bonds or the transfer of electrons in ionic bonds), hydrogen bonds are much weaker, as they only rely electromagnetic attraction.
I believe that hydrogen bonds are weaker than ionic and covalent bonds because a hydrogen bond occurs when a slightly negative molecule and a slightly positive molecule "attract" each other. Therefore, it is easier for the bond to break. On the other hand, in a covalent bond, electrons are "shared" between two atoms/molecules and in an ionic bond, there is a strong force created do to cations and anions interacting together.
Hydrogen bonds, from what I remember from high school chem, were the strongest intermolecular forces, but bonds are technically between molecules, which would explain why they're actually bonds like someone had mentioned earlier.
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