Tim Donovan wrote-
T-rex evolved great size, super-powerful jaws and robust teeth capable of
penetrating armor. These aspects suggest the archpredator coped with large,
armored prey such as Alamosaurus; otherwise they seem superfluous. Bladelike
teeth probably would have always sufficed against unarmored hadrosaurs and
even ceratopsids. Ankylosaurs alone were unlikely to have spurred the advent
of T-rex, or such theropods would have appeared much earlier. In contrast,
titanosaurs predated T-rex by only a stage or so, in Cordillera.
First of all, there is no proof Alamosaurus was armored (even assuming it is a valid genus), although I suppose the evidence is mounting that all titanosaurians were armored. Still, this armor is very sparse and I might expect a predator to go for an unarmored section, like the neck. Also, the robust teeth could have many other explanations besides penetrating armor, such as crushing bones, so that more flesh could be eaten or carcasses could be eaten easier. After all, you don't see hyaenas evolving their robust teeth to bust armor. Perhaps the appearence of large ankylosaurs significantly earlier in the fossil record (Barremian, Early Cretaceous) would be evidence suggesting that Tyrannosaurus did not evolve robust teeth for armor penetration, as you would expect robust-toothed theropods to appear alongside large ankylosaurs. However, only narrow toothed large theropods such as Utahraptor and Acrocanthosaurus are known from such deposits. Similarly, other environments with armored titanosaurs (Maevarano Formation, Madagascar; Lecho Formation, Allen Formation, Argentina) don't show any tendancy to evolve robust-toothed predators, as slender-toothed abelisaurids and carcharodontosaurids are the top predators in these locales.
As Starkov noted,
ceratopsids had to become larger to survive T-rex. The replacement of
Euoplocephalus with Ankylosaurus, which he neglected to mention, is another
example of T-rex-driven size increase.
Perhaps this happened the other way around, with Tyrannosaurus evolving it's large size and robust skull to kill the larger ornithischian species that were evolving. After all, large ornithischians are absent in Gondwanan deposits populated by abelisaurids, while both the Nemegt and Hell Creek had large ornithischians. For instance, the Nemegt was home to Tarchia, the largest Asian ankylosaur, and Shantungosaurus, one of the largest hadrosaurids. Also, the large segnosaur Therizinosaurus, which is for all intents and purposes a large unarmored herbivore, inhabited the Nemegt. Tyrannosaurus might have actually been limited to environments with large ornithischians, so that it was never able to invade South America after the land bridge formed.
If T-rex was well-adapted to hunt
titanosaurs, the results could have been devastating if or when it gained
access to the titanosaur-dominated faunas of Gondwana. The sauropods and
small ankylosaurs, etc. of southern regions had long confronted only the
relatively small and weak abelisaurs. Lacking co-evolutionary preparation to
withstand T-rex, Gondawana prey could have succumbed very quickly, just as
avian quarry did when the efficient brown tree snake entered Guam.
This lacks any material evidence. No tyrannosaurids have ever been found in South America, Cretaceous Europe, Africa or India. The abelisaurids and carcharodontosaurids of Gondwana, while having weaker jaws, were oftentimes quite large. Also, the many small-medium ornithischians of Hell Creek and Nemegt seemed to have survived fine alongside Tyrannosaurus, so I don't see why their Gongwanan equivalents would be wiped out by invading tyrannosaurs. Then there's the fact large titanosaurs lived in both the Hell Creek and Nemegt (Opisthocoelicaudia) alongside Tyrannosaurus anyway...