PC player here, but it should work the same.
A turret has 3 different modes:
Fire at will: Engages hostile targets at random depending on threat level of the targets around you, so if one NPC is judged more dangerous, your turrets will switch over autonomously
Target only: engages only your locked target and tracks this one until destruction or until you lock on to another target.
Forward fire: auto-tracking is disabled and it fires like a fixed weapon and only upon trigger press.
Check your ship function panels for "turret mode" and change it accordingly.