It doesn't matter whether you're starting a turn, or partway through it - the effect is the same. The faster you're moving, the greater the water pressure differential at the rudder, and hence the more energy provided to turn the boat. As far as I can tell - I'm no expert here on the exact physics- this is pretty accurately modelled in SH3, and also was in SH2.
Downsides: (1) of course, by going faster, you're making more noise, and (2) the mechanics of turning the boat actually slow you down somewhat for the same reason that travelling fast speeds your turn.
It works the same way for the dive-planes too. The faster you're travelling, the more effective the dive-planes are. That's why, when you order a crash dive, the Chief orders ahead flank.
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