As part of 3GPP standardization work for Release 17, non-terrestrial networks (NTNs) aim to bring 5G New Radio (NR) communications to unserved and isolated areas. Constellations of low Earth orbit (LEO) satellites have emerged as a promising asset for NTNs and a key enabler technology to provide truly seamless and ubiquitous connectivity through 5G. Varying and longer propagation delays compared to terrestrial networks, limited radio link budget and the inherent high-speed movement of LEO satellites introduce new challenges in the mobility management procedures. To guarantee robust service continuity and satisfactory user experience, the handover (HO) procedure in LEO satellite systems is critical. Motivated by this fact, this paper presents a first performance analysis of the conventional 5G NR HO algorithm in a LEO-based NTN deployment. We provide system-level simulations obtained for different values of HO margin and time-to-trigger. Furthermore, we compare the HO performance in NTN with two 3GPP terrestrial scenarios - urban macro and high-speed train. The simulation results show HO failures and radio link failures are a factor 10 higher for the NTN scenario, while the corresponding time in outage is 5 times longer. Finally, we analyze the key issues and suggest potential mobility enhancements