Well, the birds and pterosaurs really didn't overlap much in terms of size and niche, but it would make sense that even early birds could push similar-sized pterosaurs to the edge of extinction, unless the pterosaurs became specialized like the Anurognathids. There are fewer and fewer of the little ptero's as the Cretaceous progresses and birds establish themselves, which could have happened quite early. Look at *Nemicolopterus* for a possible example of this.
...if *Nemicolopterus* isn't actually a baby *Sinopterus*, that is.And what really small adult pterosaurs are there known from the Jurassic? *Pterodactylus* grew up to reach a wingspan of 2.5 m, *Rhamphorhynchus* was similar. All I can think of are the anurognathids, and those survived into the Early Cretaceous quite well.
What is more, anurognathids are _only_ known from the few sites with the most exceptional preservation. No such sites are known from the Late Cretaceous at all. We have no idea when between the Jiufotang Formation (Aptian) and Messel or the Green River Formation (early to middle Eocene) the anurognathids went extinct.