Quarks don't exist? Is that just to get a rise out of me or are you really that detached from science? That quark was originally postulated to explain the huge number of hadrons coming out of particle experiments in the 50's and early 60's. The number of hadrons and emerging patterns in properties gave a hint that there was an underlying structure. In the same way that the periodic table hints at the substructure of atoms. Deep inelastic scattering experiments were the first to prove directly that the proton was not fundamental, in the same way that nuclear scattering coule be used to show the atom had sub structure. Although so far we have only shown that the proton and the neutron are mode of something smaller not that the current theory of quarks is true. Infact while there was still a gap between the emerging theoretical opinion that quarks best fitted the data and the current experimental status the components of protons and neutrons were called partons, a term that has actually stuck today for some experimental constants largely independent of the underlying theory.
The quark hypothesis predicted a lot of heavier undiscovered particles, these became the object of searches to prove the existence of quarks. The discovery of the j/psi (two groups found it at the same time and gave it different names; they never resolved their differences hence it still having two names to this day) confirmed the existence of the charm quark giving the quark theory massive experimental backing. This is effectively the confirming experiment. However Kobayashi and Maskawa theorized another generation of quarks to explain CP violation. So they hypothesized the existence of two new quarks, named bottom and top. The bottom quark was found a few years after they made their prediction. However the top quark was not found creating a slight uncertainty in the theory. Finally the top quark was created directly at the Tevatron in 1995 it had been so hard to find because it was incredibly heavy (roughly, 200 times heavier than a proton). Now B mesons, pairs of B quarks, are created regularly to study CP violation and matter-antimatter asymmetry. Finally many of the conundrums of the atom, such as why lots of positively charged protons stay closely bound, is very precisely explained by the strong force, which exists between quarks.