C680 First Officer reported descending below their assigned altitude for traffic avoidance; resulting in flying towards terrain during initial approach. ATC issued a low altitude alert; and the flight crew climbed to a safe altitude to continued the approach.

Narrative

While on the ZZZZZ arrival into ZZZ we experienced continuous turbulence with a few pockets of severe turbulence. After being taken off the arrival for vectors into the visual approach for XXR; we noticed potentially conflicting traffic and began preparing for a potential RA. While we never received an RA annunciation; the ZZZ Approach Controller issued instructions to avoid traffic. After clearing this conflict; the controller also issued a low altitude warning. We stabilized control of the aircraft and complied with ATC instructions to stay above the Maximum Authorized Altitude (MAA) for the area; align with the final approach course; and safely configured the aircraft for a stabilized approach into ZZZ. VMC conditions throughout; no RA warnings; no GPWS warnings.30 min from arriving at our destination; my partner and I briefed our threats for the area. We made an emphasis on congested traffic in the area and potential for turbulence. We also dual verified and briefed our plan for the arrival and approach into ZZZ; with an emphasis on the MAA for the area. The slew of events began with experiencing continuous moderate turbulence with a few pockets of severe on the arrival. ZZZ Approach broke us off the arrival and issued instructions for a new heading and altitude. While complying with this clearance we became fixated on the threat of a potential RA. After being queried by ATC on our cleared altitude; we recognized it was set incorrectly. Due to the stress of the conditions; I'm unsure of when or how the AAD (Altitude Alerter Display) was set incorrectly. After we cleared the conflict; we were able to safely configure and stabilize the aircraft for the approach despite the conditions.