I guess I have a different opinion. I have no trouble using the
compass even in light turbulence. Moderate turbulence is a different
matter. Perhaps this is because the airplanes at my FBO have bad DGs
that need to be reset every 5 minutes, and we just got used to reading
the compass in bumpy air. Regarding timed turns, they will only get
you to the approximate heading. For example, even if you are only 5%
off from a standard rate turn (which is hard to tell on the TC), you
will be about 10 degrees off after a 180-turn. In order to fine tune
that heading, one needs to know about compass errors. I've seen
students zig zag their way along a north heading because they didn't
understand how to compensate for the banking errors.

I fly and teach partial panel using timed turns for a heading change of 60
degrees (20 seconds) or less, and the compass for longer turns. In both
cases, but especially with the compass, you can expect to fine tune the
heading with a second timed turn. One big mistake I see with both timed and
compass turns is that the pilot looks at the compass too soon after the turn.
I teach my students that after they begin the roll out, based on either time
or compass lead/lag, they should ignore the compass completely for the next
few seconds. The main priority after roll out is to stabilize the airplane
using the turn coordinator and VSI. After that, check the altimeter, then go
back to the TC and VSI. This allows you to verify that pitch is OK, and also
gives the compass the time it needs to settle down. Students who "zig zag
their way along a north heading" do so not because they fail to compensate for
compass errors, but because they don't give the compass time to settle down
after turning.

One item on the Instrument Rating Practical Test Standards that's often
overlooked is the turn coordinator calibration (Area IV, Task E4). People
who've never done this should try it sometime. You might be surprised at how
far off the instrument can be.

I have one other comment, not directly relevant to the compass question. All
pilots should be careful to use the correct descriptions of turbulence given
in the AIM (para.7-1-23). Moderate turbulence is pretty bad:

"Unsecured objects are dislodged."

In my experience a lot of GA pilots report Moderate turbulence when it's
really just Light ("Unsecured objects are displaced slightly"). I usually
keep my kneeboard loose on my lap, not strapped down. If it doesn't leave my
lap, the turbulence is no worse than Light.

Barry