Ultraviolet communication (UVC) is a promising technology due to its ability to operate in non-line-of-sight (NLOS) mode thereby eliminating the pointing acquisition and tracking (PAT) requirement as needed by infrared and visible light communications. However, NLOS UVC suffers from very high attenuation and turbulence-induced fading when operated over a long distance. Due to these limitations, the existing literature on the NLOS UVC is mostly restricted to short-distance communications only. Therefore, this paper addresses these challenges by proposing an outdoor subcarrier-intensity-modulation (SIM) based multi-relay cooperative communication system employing the best relay selection and decode-and-forward (DF) relaying protocol. The turbulence-induced fading is modelled using lognormal distribution under weak atmospheric turbulence conditions. We derive novel closed-form analytical expressions of outage probability and ergodic capacity. Correctness of the derived analytical expressions is validated through numerical simulations.