The layered ternary compound TaIrTe$_4$ has been predicted to be a type-II Weyl semimetal with only four Weyl points just above the Fermi energy. Performing magnetotransport measurements on this material we find that the resistivity does not saturate for fields up to 70 T and follows a $ rho sim B^{1.5}$ dependence. Angular-dependent de Haas-van Alphen (dHvA) measurements reveal four distinct frequencies. Analyzing these magnetic quantum oscillations by use of density functional theory (DFT) calculations we establish that in TaIrTe$_4$ the Weyl points are located merely $sim$ 40-50 meV above the chemical potential, suggesting that the chemical potential can be tuned into the four Weyl nodes by moderate chemistry or external pressure, maximizing their chiral effects on electronic and magnetotransport properties.