A large bird-type animal that moves at great speed, but across the ground, which I think is desert and the creature is enclosed inside a walled region. I must have read it about 40 years ago. Might it be Michael Moorcock or Jack Vance? This creature haunts my subconscious even now. And no, it's not The Roadrunner!
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You're thinking of the sandgorgon from "White Gold Wielder" in the Chronicles of Thomas Covenant. It's confined by a curse to a particular geographic region. It can leave only if someone speaks its name. Covenant is in prison in a walled city in the desert. He speaks the name; the creature comes at tremendous speed and bashes through several layers of walls to enter his cell. The curse includes pummeling the speaker to a dead pulp, but because Covenant is the 'white gold wielder' he can override the curse.
This reminds me of the sandgorgon from Stephen R. Donaldson's various and sundry works about Thomas Covenant. It's been… uh… decades since I last read the first two series, but as I recall, the sandgorgons—elementally fierce two-legged creatures from the desert—are only mentioned in the first trilogy, but one is encountered in the The One Tree (the second book of the second trilogy), and has yet another, perhaps deeper, encounter in White Gold Wielder, which concludes that trilogy.
Points about sandgorgons that match your memory:
They have a backward-bending (digitigrade) leg.
They are described as moving phenomenally fast. For example, the character Covenant summons one across intercontinental distance in a time measured by minutes merely by calling its name.
Sandgorgons were historically imprisoned in the wizard "Kasren's Gyre," which was a massive magical maze-like sandstorm that equaled their ferocity. I believe that the last four books (which I have not read, so grain of salt) address the migration and consequences of the sandgorgons leaving the captivity of the Gyre.
The creatures are also described as being bleached by the sun.
How about The Iron Thorn by Algis Budrys? That has the Amsirs which are flightless birds. More info here
Maybe Hyperion? The creature sounds like The Shrike...