Summary from Baron 2021:
“the clade Pterosauria belongs with Lagerpetidae as part of a broader Pterosauromorpha that then, with Dinosauriformes, falls within Ornithodira.”
And for illustrations
Baron republishes images of Dimorphodon from 1870.
From the abstract
“Our understanding of the pterosaurs’ place within the reptilian lineage has had a long and complex history. The unusual morphology of pterosaurs, which is inextricably linked to their habit of powered flight, has generated numerous proposals over the years regarding their exact origin and systematic position. Though it was concluded early on in pterosaur research history that these animals represented a group of derived flying reptiles, their exact origination remained mysterious for a long time and is still somewhat controversial. A rough consensus has now been reached that pterosaurs are derived archosaurs and are likely close relatives of the dinosaurs, united with them in the clade Ornithodira, though some still challenge this view. The anatomical evidence in support of this position close to Dinosauria is also admittedly fairly limited at present, largely owing to a lack of any clear-cut transitional ‘proto-pterosaur’ taxa (albeit that some fragmentary specimens have been suggested to represent exactly this). Differing hypotheses have also recently been put forward as to the exact interrelationships between the pterosaurs and other non-dinosaurian and dinosaurian ornithodirans. Here the previous hypotheses of where pterosaurs fit into the reptilian lineage and the anatomical evidence in support of the current hypotheses are reviewed. Results of new analyses are included that looked to test the origin and systematic position of the Pterosauria using an expanded version of a large anatomical dataset of archosaurs, within which several previously unconsidered early pterosaur taxa and a suit of new anatomical characters were considered. The analyses in this study support the close affinities between pterosaurs and dinosauriforms within Ornithodira; Pterosauria is recovered as the sister-taxon to Lagerpetidae. Such a result suggests that the clade Pterosauria belongs with Lagerpetidae as part of a broader Pterosauromorpha that then, with Dinosauriformes, falls within Ornithodira. The anatomical evidence in support of this position within Ornithodira is also discussed in detail.”
This is my reply, after receiving the PDF from author Matthew Baron.
Thank you, Matthew. And, of course, I will not share this PDF.
Some uncited research that supports the fenestrasaur hypothesis:
http://davidpetersstudio.com/papers.htm
Peters 2000 was my first paper and as a naive freshman, I made several errors, including the assignment of the tanystropheids to the prolacertiformes, corrected in Peters 2007 and in later works listed above and below.
http://reptileevolution.com/reptile-tree.htmhttp://reptileevolution.com/cosesaurus.htmhttp://reptileevolution.com/sharovipteryx.htmhttp://reptileevolution.com/longisquama.htmhttp://reptileevolution.com/MPUM6009.htm
There is a movement to keep Langobardisaurus, Cosesaurus, Sharovipteryx and Longisquama out of pterosaur origin cladograms, and you followed it. I can’t understand the motivation for doing so. Please share with me when you have an opportunity.
re: My freshman attempt (Peters 2000): Here is a link to a more mature researchgate.net paper the referees did not want published.
https://www.researchgate.net/publication/328388115_Cosesaurus_aviceps_Sharovipteryx_mirabilis_and_Longisquama_insignis_Reinterpreted
You can’t shed light on the origin of pterosaurs by deleting pterosaur precursors.
Give yourself time to see the pertinent fossils, as I have. Cosesaurus has a prepubis, a pteroid, a five vertebrae sacrum, elongated ilium, a precursor sternal complex (completed in Longiquama) and have locked-down elongate coracoids, a trait shared only with flapping tetrapods (bats substitute a long clavicle). See link above.
Finally, remind yourself that a small Tanystropheus was originally and mistakenly considered a pterosaur because it had dimorphic teeth and pterosaur-like feet (that metapodial fifth toe is what attracted my attention to Cosesaurus in the first place).
Thank you for the one citation. Sorry you missed the others.
Don’t borrow cladograms and reconstructions traced from fossils. Step up to the plate and create your own, as I have.
Best regards, and better luck next time.
David Peters
Darren Naish often accuses me of ‘bad methodology’.
In this case (like Hone and Benton 2007, 2008), Matthew Baron 2021 did not see the pertinent fossils nor did he include the pertinent fossils in his borrowed cladogram. Is that what Naish would consider ‘good methodology’? It will be interesting to see if Naish reserves his accusations to colleagues and spends them only on outsiders.
References
Baron MG 2021. The origin of pterosaurs. Earth-Science Reviews 221: 103777
https://doi.org/10.1016/j.earscirev.2021.103777
Hone DWE and Benton MJ 2007. An evaluation of the phylogenetic relationships of the pterosaurs to the archosauromorph reptiles. Journal of Systematic Palaeontology 5:465–469.
Hone DWE and Benton MJ 2008. Contrasting supertree and total evidence methods: the origin of the pterosaurs. Zitteliana B28:35–60.
The Pterosaur Heresies 
Quick note here: BaRon, not BaCon. It’s misspelled twice in the post. Aside from that, I and others agree that adding Longisquama and Sharovipteryx to these cladograms would go a long way to sussing out pterosaur origins.
Many thanks for the spell check. A person’s name is important.