Was incest all that common amongst royalty during the Middle Ages?

Depends on what you mean by incest! If you mean sibling incest, parent-child incest, incest between aunts and uncles and their nieces and nephews, etc. this was very very uncommon and socially unacceptable. Even in fiction, it was a topic that elicited horror, and its depiction there seems to have haunted the Middle Ages; familial incest such as fornication between brother and sister or father and daughter was a sin as well as a crime. When it was found to have occurred among commoners, it carried penalties both secular and religious, and might be punished by prison, exile, or death. When it even reputed to occur among the upper classes (think the Borgia family) it was a source of scandal. There were no traditions of Ancient Egyptian-style sibling marriage, and for royal prospective brides and grooms the more important concern was often securing an advantageous link with another country and another dynasty -- for instance, the parties responsible for securing a suitable match for a 14th century English monarch or prince/princess of marrying age would be hunting for suitable candidates in France, Spain, and the Holy Roman Empire rather than the lucky bride or groom's own cousins. Any candidates from one's own country would still have their pedigree thoroughly examined for possible advantages, and any degree of affinity would need to be known to prevent legal trouble later along the line. The levels of unacceptable closeness between spouses or sexual partners were first laid out, as far as the medieval Church was concerned, in chapter 18 of Leviticus -- you can't sleep with your mom, you can't sleep with your father's wife, you can't sleep with your sister whether she was raised alongside you or not, you can't sleep with your niece, you can't sleep with your aunt, you can't sleep with your daughter-in-law. Having sex with your spouse is an integral part of marriage, therefore, you can't marry any of those women either, even if they're widowed. Even if actual practice may have varied, this passage is central to how medieval Christians (and likely members of other Abrahamic religions, I just can't vouch for those) have understood incest.

In the early Church, the horror of even accidental incest was such that you get polemic writers using the repulsiveness of it as a disincentive to fornication as well as child abandonment -- sleep with a prostitute and you might be sleeping with your own child unawares, one you abandoned a decade or two ago and promptly forgot about. In the early Middle Ages, the Church was very serious about the whole range of people you couldn't marry, and there was a lot of wrangling about the concept of consanguineous marriage in canon law, marriage within bounds of relatedness that were deemed too close and thus prohibited. These bounds got built up somewhat from the premises outlined in Leviticus. In the 8th century Pope Gregory II forbade even marriage to relatives by godparenthood -- the godmother of your own child, for instance, or the mother of the child to whom you were godfather. By this pre-Lateran model of canon law, first, second, third, fourth, fifth, etc. cousins were all out of bounds, as were the wives and husbands of dead aunts and uncles -- in general marriages were prohibited between individuals within seven degrees of affinity to each other, and affinity could include relatives by marriage as well as blood. But the problems with this become obvious, and too many important people needing to request papal dispensations on that basis led to revamping and relaxation of these policies at the Fourth Lateran Council in 1215 and then later at the mid-16th century Council of Trent -- relaxed, but not so relaxed that you could marry your sister, or your aunt, or even your dead brother's wife. That last one caused massive headaches for Henry VIII and his allies first in 1509 and then in 1525, well before the Council of Trent -- the king's first wife, Catherine of Aragon had first been the wife of Henry's older brother Arthur. Her marriage to Henry after Arthur's death required both a papal dispensation and a plausible case to be made that Arthur and Catherine had never consummated their marriage. That way their marriage could be retroactively annulled due to lack of consummation, thus severing the tie that would make Henry's marriage to Catherine incestuous, and Henry could do whatever he wanted with the lovely Catherine with impunity; she'd never "really" been his brother's wife to start with. But then some sixteen years later, when Henry wanted to end his marriage to Catherine in order to marry Anne Boleyn, things go pear-shaped all over again. There are other instances of marriage complicated by the standard degrees of affinity, such as uncle-niece incest, but they're all complicated by these things and not facilitated by them. To marry within even ambiguous degrees of affinity was to put your marriage in peril.

So dispensations were possible for nobility, but they could be tough or easy to get depending on how much leverage you had and how much the current Pope liked you. Nevertheless, being known to schtup your cousins (or, God forbid, your siblings) was not desirable for a whole host of reasons; there would be consequences for fornication if you were unmarried and for the legal state of your marriage if you were married, and in general it wasn't something you wanted to get tangled up in. But if you mean inbreeding, what we think of when we think of the Habsburgs and their Habsburg jaws, this was a product of royal intermarriage that became especially pronounced as a late medieval and Early Modern phenomenon. Over time -- that is, over centuries, not decades -- the pools of potential marriage partners were narrowed by political and religious affiliations, and it became harder and harder to find other individuals of suitable rank to whom you weren't related in some way. This wouldn't be nearly as big of a problem for common people, but for any individual with a title, keeping track of your family tree was vital and effectively everyone involved knew who was related to whom and how closely/by which relatives. This is where we start to get the idea of inbred deformed royalty -- not from a single progenitor marriage between a brother and sister, but through centuries of family trees of European royal houses becoming increasingly interrelated.

So in short, incest was only common among medieval royalty when "incest" was defined most broadly. Sexual abuse and assault were realities of medieval life, then as now and likely across boundaries of class, but incestuous marriage and fornication between members of the same nuclear family was no more encouraged in medieval Europe than it is now.