The Zwicky Transient Facility (ZTF) has discovered five events (0.01 < z < 0.4) belonging to an emerging class of active galactic nuclei (AGNs) undergoing smooth, large-amplitude, and rapidly rising flares. This sample consists of several transients initially classified as supernovae with narrow spectral lines. However, upon closer inspection, all of the host galaxies display Balmer lines with FWHM(H beta) similar to 900-1400 km s(-1), characteristic of a narrow-line Seyfert 1 (NLSy1) galaxy. The transient events are long lived, over 400 days on average in the observed frame. We report UV and X-ray follow-up of the flares and observe persistent UV emission, with two of the five transients detected with luminous X-ray emission, ruling out a supernova interpretation. We compare the properties of this sample to previously reported flaring NLSy1 galaxies and find that they fall into three spectroscopic categories: 1) Balmer line profiles and Fe ii complexes typical of NLSy1s, 2) strong He ii profiles, and 3) He ii profiles including Bowen fluorescence features. The latter are members of the growing class of AGN flares attributed to enhanced accretion reported by Trakhtenbrot et al. We consider physical interpretations in the context of related transients from the literature. For example, two of the sources show high-amplitude rebrightening in the optical, ruling out a simple tidal disruption event scenario for those transients. We conclude that three of the sample belong to the Trakhtenbrot et al. class and two are tidal disruption events in NLSy1s. We also hypothesize as to why NLSy1s are preferentially the sites of such rapid enhanced flaring activity.