Most plants don't exactly turn towards the sun (except sunflower heads,
and a few others) - they grow towards it. Not all of phototropism (growing
towards light) is understood yet.
What I remember from my botany degree is that Darwin gave us part of the
answer in 1880 when he showed that the tip of a plant turns towards the
strongest source of light by growing faster on the dark side. This has
since been shown to be due to less release of a plant growth hormone
called auxin (Indole Acetic Acid) on the light side than on the other.
Result: more cell division and growth on the darker side, at least until
the tip is growing straight towards the light. However, it is known that
more than just auxin is involved in plant phototropism...
Another response of plant growth to light is the seedling elongation which
continues until the cotyledon(s) are exposed to light (something about a
compound within the cotyledon shifting from one form to another in
response to red/ far red light).
So do plants grow faster in the dark (or on the dark side) ? Maybe, but
only to get to more light. :)