How are attitude indicators kept accurate?
A gyroscopic AI has an erection mechanism, which continuously corrects the AI to be upright based on the local level, or downward acceleration vector. The correction rate is generally 3-5 degrees per minute.

The way that the AI corrects itself is a system of pendulous vanes. When the gyro is not upright relative to the local level, centrifugal force pushes open vanes on the gyro's case. Air escapes through the uncovered holes, applying a force to the case, and correcting for precession.


(FAA AC 65-15A)

The second half is somewhat counter-intuitive, but flying in a coordinated turn will not continuously increase the error in the AI. For the first half of the turn, the AI will add error; in the second half, it removes the error. After a 360 degree turn, the precession error will be removed completely. For more detail, see another answer: If I fly a balanced turn long enough, will the AI show I'm level?