In 2011,
back when the large reptile tree (LRT, 1786+ taxa) was much smaller (only about 400 taxa) I attempted a rather odd test. I wondered if turtles and pterosaurs (both Lepidosauromorpha in the LRT) would nest together while playing taxon deletion games. Here’s the 2011 link:
https://pterosaurheresies.wordpress.com/2011/07/25/pterosaurs-and-turtles-say-it-aint-so/
Click here to see those 2011 turtle-pterosaur results, still posted online.
Other workers interested in pterosaurs
(most recently Ezcurra et al. 2020) also like to play taxon deletion games as they attempt to cherry-pick preferred sisters close to dinosaurs while omitting tested and validate sisters far from dinosaurs.
The backstory
Peters 2000 added four tritosaur tanystropheid, fenestrasaur pterosaur precursors, Langobardisaurus, Cosesaurus, Sharovipteryx, and Longisquama (Fig. 1) to four previously published analyses and in every case these four nested closer to pterosaurs than any archosaur, archosauriform or archosauromorph. Unfortunately those taxa were omitted from more recent analysis, like those of Kellner 2003, Unwin 2003, Hone and Benton 2007, 2008, Bennett 2012 and Ezcurra et al. 2020.
A few years later, but still 14 years ago,
Peters 2007 added the lepidosaur, Huehuecuertzpalli and it attracted the four fenestrasaurs + pterosaurs. The LRT nested turtles within the Lepidosauromorpha here in 2011, updated here in 2014.
Now that many more taxa are present in the LRT,
let’s rerun that test and its various deletion subunits.
Today, in 2020, repeating the experiment with more taxa
deleting all lepidosauromorphs, other than turtles (and their ancestors back to Stephanospondylus) and pterosaurs, and keeping all archosauromorphs and enaliosaurs. Outgroups retained = Gephyrostegus and Silvanerpeton.
Results: Pterosaurs nest with turtles and basal sea reptiles rather than archosaurs and archosauromorphs.
Adding back all basal diapsids and protosaurs
Results: Basal diapsids as the first large clade, followed by protorosaurs with tritosaurs based on the convergence found there.
Adding back all tritosaurs
(= Macrocnemus as the last common ancestor) nests turtles and pareiasaurs as the first large clade, tritosaurs (including pterosaurs) as the next large clade, followed by archosauromorophs (including Lagerpeton and Scleromochlus).

Figure 1. Squamates, tritosaurs and fenestrasaurs in the phylogenetic lineage preceding the origin of the Pterosauria.
Deleting all non-fenestrasaur tritosaurs
(= Cosesaurus as the last common ancestor)
Results: Fenestrasaurs nest with basal diapsids. Orovenator is the proximal outgroup.
Deleting all non-pterosaur fenestrasaurs
(Bergamodactylus as the last common ancestor).
Results: Pterosaurs nest between turtles and choristoderes.

Figure 2. Basal diasids and proto-diapsids. Largely ignored these putative synapsids actually split from other synapsids while retaining the temporal fenestra trait that serves as the basis for the addition of upper temporal fenestra in diapsids. Included here are Protorothyris, Archaeovenator, Mycterosaurus, Heleosaurus, Mesenosaurus, Broomia, Milleropsis, Eudibamus, Petrolacosaurus, Spinoaequalis, and Tangasaurus.
Deleting all basal diapsids
(only turtles, pterosaurs and archosauromorphs are ingroup taxa).
Results: Pterosaurs nest between turtles and choristoderes, far from Scleromochlus, dinosaurs and Lagerpeton.
Deleting all turtle ancestors
(= deleting Stephanospondylus through pareiasaurs)
Results: Pterosaurs nest between turtles and choristoderes, far from Scleromochlus, dinosaurs and Lagerpeton.

Figure 3. Faxinalipterus matched to Scleromochlus. The former is more primitive, like Gracilisuchus, in having shorter hind limbs and more robust fore limbs. The maxilla with fenestra and fossa, plus the teeth, are a good match.
Re-inserting terrestrial younginiforms and protorosaurs
to this last taxon list.
Results: Two large clades follow the turtle clade. Pterosaurs nest between three basal Youngina specimens and the clade Protorosauria, apart from the terrestrial younginiformes (other Youngina specimens + Pararchosauriformes (= Proterosuchus as the last common ancestor and choristoderes) and Euarchosauriformes (= Euparkeria as the last common ancestor and Lagerpeton and Scleromochlus).

Figure 4. The closest kin of Tropidosuchus are the much larger Chanaresuchus (matching Nesbitt 2011) and the smaller Lagerpeton.
Bottom line:
With or without tritosaurs and fenestrasaurs, pterosaurs prefer to nest with terrestrial younginiforms, choristoderes or turtles rather than lagerpetids, dinosaurs or Scleromochlus. Taxon exclusion remains the problem in traditional cladograms (like the recent Ezcurra et al. 2020).
Please send this post to anyone who still believes or protects
the outmoded clades ‘Ornithodira’ or ‘Avemetatarsalia’. Too many professors and their students are clinging to invalidated myths based on taxon exclusion — which is not what real scientists do. Real scientists test all competing candidates without cherry-picking or omitting taxa to suit their personal whims and traditions, in fear of their professors or colleagues.
If you would like to play taxon deletion games with the LRT,
click here, then click on the yellow CLICK HERE for LRT MacClade.nex file box with your request.
References
Bennett SC 1996. The phylogenetic position of the Pterosauria within the Archosauromorpha. Zoological Journal of the Linnean Society 118:261-308.
Bennett SC 2012. The phylogenetic position of the Pterosauria within the Archosauromorpha re-examined. Historical Biology. iFirst article, 2012, 1–19.
Benton MJ 1983. The Triassic reptile Hyperodapedon from Elgin, functional morphology and relationships. Philosophical Transactions of the Royal Society of London, Series B, 302, 605-717.
Brusatte SL , Benton MJ , Desojo JB and Langer MC 2010. The higher-level phylogeny of Archosauria (Tetrapoda: Diapsida), Journal of Systematic Palaeontology, 8:1, 3-47.
Evans SE 1988. The early history and relationships of the Diapsida. In: M. J. BENTON (Ed.), The Phylogeny and Classificationof the Tetrapoda. 1. Amphibians, Reptiles, Birds. Systematics Symposium Association Special Volume; Oxford (Clarendon Press), 221–260.
Ezcurra MD et al. (17 co-authors) 2020. Enigmatic dinosaur precursors bridge the gap to the origin of Pterosauria. Nature (2020). https://doi.org/10.1038/s41586-020-3011-4
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.
Jalil N-E 1997. A new prolacertiform diapsid from the Triassic of North Africa and the interrelationships of the Prolacertiformes. Journal of Vertebrate Paleontology 17(3), 506-525.
Kellner AWA 2003. Pterosaur phylogeny and comments on the evolutionary history of the group. Geological Society Special Publications 217: 105-137.
Nesbitt SJ 2011. The early evolution of archosaurs: relationships and the origin of major clades. Bulletin of the American Museum of Natural History 352: 292 pp.
Peters D 2000b. A Redescription of Four Prolacertiform Genera and Implications for Pterosaur Phylogenesis. Rivista Italiana di Paleontologia e Stratigrafia 106 (3): 293–336.
Peters D 2000. A Redescription of Four Prolacertiform Genera and Implications for Pterosaur Phylogenesis. Rivista Italiana di Paleontologia e Stratigrafia 106 (3): 293–336.
Peters D 2007. The origin and radiation of the Pterosauria. In D. Hone ed. Flugsaurier. The Wellnhofer pterosaur meeting, 2007, Munich, Germany. p. 27.
Sereno PC 1991. Basal archosaurs: phylogenetic relationships and functional implications. Journal of Vertebrate Paleontology 11 (Supplement) Memoire 2: 1–53.
Unwin DM 2003. On the phylogeny and evolutionary history of pterosaurs. Pp. 139-190. in Buffetaut, E. & Mazin, J.-M., (eds.) (2003). Evolution and Palaeobiology of Pterosaurs. Geological Society of London, Special Publications 217, London, 1-347.