Energy is not released when bonds are broken. Energy is released when bonds form. It takes energy to break any bond. Otherwise it would not be a bond.
Here's what energy of a chemical bond looks like versus internuclear distance.
As the bond forms, electrons move into the internuclear distance, a bonding orbital. The negative charge density between the atoms within the bonding orbital holds the nuclei together. The entire system has fallen into a potential energy well.
A strong, stable bond is one that has a deep potential energy well, while a weak, unstable bond has a shallower well, higher potential energy, but still below the energy of separate neutral atoms.
Chemistry is about describing the transformations that matter undergoes. Where does the energy come from that is released when bonds are broken? The energy doesn't derive from the broken bonds. The energy derives when the new bonds form in the product. Remember, in chemistry, you aren't only breaking apart A, you are also forming B. High energy bonds are weak bonds because after the energy is input to break them, the atoms are available to form newer, stronger, lower energy bonds in the product.
Imagine the reaction of glucose and oxygen. Imagine first pulling all of those bonds apart, lifting the atoms out of the potential energy wells of the original bonds. At that point, there would only be the input of energy. Now, let the atoms fall together to form the product, carbon dioxide and water. The carbon-oxygen and oxygen-hydrogen bonds in the product are deeper potential energy wells than the original bonds in oxygen molecule and glucose. This is largely because oxygen, being so electronegative, can pull the bonding electrons in so close to its electron greedy nucleus in the product.
We say weak bonds are unstable because if a pathway is available to form new molecules through chemical reaction with stronger bonds, the internal energy decrease leads to exothermic process which tends to increase the entropy of the universe, so the arrow of time points away from weak bonds.