Blast Doors are designed to be immune to laser and blaster fire of this caliber. An anti-personale turret would not be able to damage the door as such. Fopr an example of a blast door, think about the doors in Episode I when Ob-Wan and Qui-Gon have to use their lightsabers to break in.
Of course the door is not invincible, and I don't doubt your ability to destroy the turrets- but getting past that door was done rather quickly.