Shortly after their inception, superconducting nanowire single-photon detectors (SNSPDs) became the leading quantum light detection technology. With the capability of detecting single-photons with near-unity efficiency, high time resolution, low dark count rate, and fast recovery time, SNSPDs outperform conventional single-photon detection techniques. However, detecting lower energy single photons (<0.8 eV) with high efficiency and low timing jitter has remained a challenge. To achieve unity internal efficiency at mid-infrared wavelengths, previous works used amorphous superconducting materials with low energy gaps at the expense of reduced time resolution (close to a nanosecond), and by operating them in complex milliKelvin (mK) dilution refrigerators. In this work, we provide an alternative approach with SNSPDs fabricated from 5 to 9.5 nm thick NbTiN superconducting films and devices operated in conventional Gifford-McMahon cryocoolers. By optimizing the superconducting film deposition process, film thickness, and nanowire design, our fiber-coupled devices achieved >70% system detection efficiency (SDE) at 2 μm and sub-15 ps timing jitter. Furthermore, detectors from the same batch demonstrated unity internal detection efficiency at 3 μm and 80% internal efficiency at 4 μm, paving the road for an efficient mid-infrared single-photon detection technology with unparalleled time resolution and without mK cooling requirements. We also systematically studied the dark count rates (DCRs) of our detectors coupled to different types of mid-infrared optical fibers and blackbody radiation filters. This offers insight into the trade-off between bandwidth and DCRs for mid-infrared SNSPDs. To conclude, this paper significantly extends the working wavelength range for SNSPDs made from polycrystalline NbTiN to 1.5-4 μm, and we expect quantum optics experiments and applications in the mid-infrared range to benefit from this far-reaching technology.