The Zwicky Transient Facility (ZTF) has discovered five new events belonging to an emerging class of AGN undergoing smooth flares with large amplitudes and rapid rise times. This sample consists of several transients that were initially classified as supernovae with narrow spectral lines. However, upon closer inspection, all of the host galaxies display resolved Balmer lines characteristic of a narrow-line Seyfert 1 (NLSy1) galaxy. The transient events are long-lived, over 400 days on average. We report UV and X-ray follow-up of the flares and observe persistent UV-bright 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: Transients with 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. (2019). 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. (2019) class, and two are TDEs in NLSy1s. We also aim to understand why NLSy1s are preferentially the sites of such rapid enhanced flaring activity.