There are numerous synonyms for the term "notorious affair" that one can use depending on the context. Some of the common alternatives include scandalous liaison, infamous fling, notorious love affair, public love scandal, incriminating relationship, improper romance, and disgraceful entanglement. Each of these substitutes conveys a different nuance and degree of negativity associated with a particular situation, be it a romantic relationship or an inappropriate conduct by a public figure. While "notorious affair" may be the most popular expression, it is always a good idea to explore other equivalents to add variety and clarity to one's writing or speech.