I'll concede that Natasha hasn't really won a fair fight but the flims do or at least try to convey that Black Widow is slightly superior to him in combat due to her having the upperhand more often than not than he did in most of their fights. (on top of her having more feats)
But whenever she gets the upperhand there's usually context, i.e. Hawkeye and Widow talking in Civil War about them still being friends when Natasha continues attacking etc. I'm OK with Widow being slightly more skilled due to more having more feats in general to showcase her skill, but IMO Hawkeye hasn't taken a clear loss against her to make her outright superior to him in terms of film portrayal.