The recent IceCube detection of TeV neutrino emission from the nearby active galaxy NGC 1068 suggests that active galactic nuclei (AGNs) could make a sizable contribution to the diffuse flux of astrophysical neutrinos. The absence of TeV γ-rays from NGC 1068 indicates neutrino production in the vicinity of the supermassive black hole, where the high radiation density leads to γ-ray attenuation. Therefore, any potential neutrino emission from similar sources is not expected to correlate with high-energy γ-rays. Disk-corona models predict neutrino emission from Seyfert galaxies to correlate with keV X-rays because they are tracers of coronal activity. Using through-going track events from the Northern Sky recorded by IceCube between 2011 and 2021, we report results from a search for individual and aggregated neutrino signals from 27 additional Seyfert galaxies that are contained in the Swift's Burst Alert Telescope AGN Spectroscopic Survey. Besides the generic single power law, we evaluate the spectra predicted by the disk-corona model assuming stochastic acceleration parameters that match the measured flux from NGC 1068. Assuming all sources to be intrinsically similar to NGC 1068, our findings constrain the collective neutrino emission from X-ray bright Seyfert galaxies in the northern sky, but, at the same time, show excesses of neutrinos that could be associated with the objects NGC 4151 and CGCG 420-015. These excesses result in a 2.7σ significance with respect to background expectations.