'gym' as in an organization where you purchase a membership, not just a gymnasium, then? Seeing 'sportschool' I assumed 'sports academy' or something similar.
Yes exactly, "sportschool" is the type with memberships, usually with a focus on fitness workouts. A gymnasium would be a "gymzaal" or a "sportzaal" (a room equipped for sports), or if you are talking about an entire building or complex for sports a "sporthal" or "sportcentrum".
In South Africa a number of our schools are known as gymnasiums as well. They certainly don't focus only on sport though!
Neither is that the case in the Netherlands. A Gymnasium is the highest and most academic category of secondary school in the Netherlands, they even teach Latin and Ancient Greek.
Thus a 'fitness centre' or 'fitness center' would be the correct translation. (I reported it.) A gym - at least in US English - is the room in a school where sports activities are exercised and contests can be held.