Leeway on the circuit pattern
I've flown at Shoreham for ten years or so and I have heard on a few rare occasions pilots departing on R20 requesting, and being granted by ATC, "an early (or immediate) left turn". I think I have occasionally also heard ATC requesting pilots to do the same thing presumably for assisting with traffic management - but I can't recall specific examples of the latter.
I have never asked for one myself and indeed I've noticed that it tends to be experienced, or confident-sounding, pilots who do and who seem to me to want to be en route east, or north east, from Shoreham as efficiently as possible.
I have sat in the right seat on one such departure where the (experienced) PIC did just this and the track took us past the tower and along the railway line and over the Adur recreation ground, as we turned and climbed eastwards.
The other thing when doing this, which makes sense, is to climb quickly because once over the River Adur (in a matter of seconds on such a departure) you are over a large and densely built up area where a decent height for safety and noise abatement is a good idea. This rational motive for a speedy climb, in my mind, is more likely than some of the the alternatives suggested in this thread.
Finally, it is very easy to envisage the track of such a departure being, for a time, precisely in line with the crosswind leg of an aircraft carrying out an accurately executed crosswind join for R20.