Proton-proton fusion happens at energies around 15 keV. The LHC currently operates at an energy of 13 TeV, which is literally one billion times larger. Fusion is one of the ***lowest-energy*** processes that could occur at the LHC, and most of the interesting reactions that are studied there go ***way beyond*** nuclear fusion.

A proton is made up of fundamental, pointlike bits of matter called quarks, held together by force carriers called gluons. These quarks are bound together with a certain _binding energy_; the quarks are also relatively light, so this binding energy actually makes up most of the mass of the proton. In other words, the binding energy of the proton is roughly equal to the mass of the proton. 

When you collide protons at an energy that's far below its mass, the proton acts as a single object. This is the regime in which nuclear physics typically occurs; all of the reactions that break nuclei apart, and the energy that holds them together, are typically small enough that it works to treat the proton as a single unit most of the time. This includes nuclear fusion; the mass of the proton is 938 MeV, and the amount of energy required for nuclear fusion (15 keV, as we said) is about a million times smaller than that.

If we instead decide to collide protons at energies that are _higher_ than a few times the proton mass, then there is now enough energy in the reaction for the quarks inside the protons to interact with each other directly. We can no longer treat protons as single objects, since the energy involved in the reaction is enough to expose its inner components. As the collision energy becomes higher and higher, the binding energy of the proton becomes less and less relevant, and the picture of two dense clouds of quarks and gluons interacting becomes more and more accurate. These reactions have enough energy to produce all manner of exotic matter, particles that you could never find in a nucleus, nor could you create them by nuclear fusion. The current energy of the LHC is roughly ten million times greater than the mass of the proton; at those energies, the proton's mass and binding energy are completely negligible, and the interesting reactions are _entirely_ driven by individual quarks and gluons interacting with each other. The proton is merely a vehicle for us to deliver bits of fundamental matter to the collision site. If there were a way to just collide free quarks and free gluons, without all the mess generated by the proton, many particle physicists would jump at the chance. (Unfortunately, this turns out to be impossible.)

There's certainly lots of extremely interesting physics to do at the nuclear-physics scale, and our understanding of the dynamics of nuclei is the subject of a great deal of active research in other experiments. But that's simply not what the proton-proton collisions at the LHC are meant to explore. The proton-proton collisions are actually better thought of as _quark-quark_ collisions, or _quark-gluon_ collisions, or _gluon-gluon_ collisions. They're not meant to study protons, which is why the protons are often totally destroyed in the collision, transformed into exotic matter which then decays back into ordinary matter. Proton-proton collisions are meant to make conclusions about the interactions of fundamental particles, and that requires acceleration to extremely high energies, far above what you would need, or want, for nuclear fusion.

So, given this, why aren't LHC physicists worried about triggering nuclear fusion? The answer is fairly straightforward: though each _individual_ proton has an energy a billion times larger than the fusion threshold, the _total amount of energy_ that is released into the surrounding area is still rather tiny, on a macroscopic scale. After all, 13 TeV is still only