Bastardy rates range from between 1-9%, depending on the historian country, and time period, so there's a bit of flexibility on how widespread it is (as you've picked GoT and Tudors, the Early Modern period is probably around 3% in England).

There's two kinds of bastardy, general and special. General bastardy meant that the parents never married, but special bastardy meant that the child was made legitimate by marriage, regardless of when that marriage took place after the birth.

How it was accepted also depends on the time period a little. Richard III contested his nephew's legitimacy (Edward V), and raised the fact in parliament that all Edward's children were bastards and could not inherit or claim "any thing by enheritance, by the law and custome of Englond". Louis XIV on the other hand, tried to make his illegitimate children legitimate in his will (the so-called Affair of the Princes), but you might want to note that in the former case, Richard might have a personal justification for his view! Henry Tudor made his bastard son the Duke of Richmond, and his uncle also was a bastard but seems to have not suffered because of it.

I can't give statistics for it because I haven't found it for royalty.