We report the analysis of all Swift observations available up to 2019 April of $gamma$-ray-emitting narrow-line Seyfert 1 galaxies (NLSy1). The distribution of X-ray luminosities (and fluxes) indicates that the jet radiation significantly contributes to their X-ray emission, with Doppler boosting making values higher than other radio-loud NLSy1. The 0.3-10 keV photon indices are on average harder with respect to radio-quiet and radio-loud NLSy1, confirming a dominant jet contribution in X-rays. However, the lower variability amplitude with respect to blazars and the softening of the spectrum in some periods suggests that also the corona radiation contributes to the X-ray emission. In optical and ultraviolet (UV) significant flux changes have been observed on daily, weekly, and monthly time-scale, providing a clear indication of the significant contribution of the jet radiation in this part of spectrum. A strong correlation between X-ray, UV, and optical emission and simultaneous flux variations have been observed in 1H 0323+342, SBS 0846+513, PMN J0948+0022 as expected in case the jet radiation is the dominant mechanism. Correlated multiband variability favours the jet-dominated scenario also in FBQS J1644+2619 and PKS 2004-447. The summed X-ray Telescope spectra of 1H 0323+342, SBS 0846+513, PMN J0948+0022, and FBQS J1644+2619 are well fitted by a broken power law with a break around 2 keV. The spectrum above 2 keV is dominated by the non-thermal emission from a beamed relativistic jet, as suggested by the hard photon index. A Seyfert-like feature like the soft X-ray excess has been observed below 2 keV, making these $gamma$-ray-emitting NLSy1 different from typical blazars.
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