Making bonds releases energy, that's why they exist. As compared with lone atoms, molecules contain less energy. If the energy reference were lone atoms, all molecules would have a negative energy.
Lone atoms are very rare where we live, so reactions rearrange atoms from molecules to other molecules. Because lone atoms would be inconvenient as an energy reference, the "heat of formation" of molecules is always referred to other molecules, like O2 or solid C for instance.
In a reaction, produced heat comes from the products, like CO2 or H2O for instance. The reactants take energy to break.
So while chemical energy does relate with chemical bonds, the strength of the bonds in the reactants reduces the heat produced by a reaction. Reactions that provide much heat, like combustions, make molecules with strong bonds as products, like CO2 or H2O, from molecules with weaker bonds as reactants, like O2 or hydrocarbons.