It's already been answered, but if you're determined to have a reptilian mount, you would be better off looking at a varanid, agamid or massive snake as a mount than a bearded dragon relative.
Willk didn't name the exact reason why the S-shaped movements limit breathing, it's called Carrier's Constraint. Varanids (goannas, perenties) use gular pumping to help force air into their lungs as they move. Gular pumping expands and constricts the throat to force air into the lungs as the monitor moves so they can maintain a relatively active lifestyle. (A paper to look up would be 'Contribution of Gular Pumping to Lung Ventillation in Monitor Lizards'. There's a free pdf of it available on google scholar)
Agamids run bipedally so avoid the constraint entirely and snakes can breathe and move at the same time (sidewinding would the preferred Sahara movement method and the serpent could likely employ similar tactics boas use to climb trees to maneuver in a Grand Canyon type setting)
Also, if you want to be really different, consider looking at the family tree of the Crocodylomorpha, there were armadillow-like crocs (Armadillosuchus) and crocs that could gallop on land (Pristichampus) to scratch at the surface.
In comparison to camels and horses, your best option is a camel really. They are completely adapted for desert living, are capable of high rates of activity regardless of the temperature swinging between hot and cold and due to their herbivorous diets would be easier to feed than a reptile (where a camel can immediately eat any available vegetation, your reptile would need to be fed on animals that are eating vegetation, in a desert life is relatively scarce so needing to feed a massive mount on primary consumers would be difficult to do in a way that would maintain more than a few such mounts at a time). And a camel beats a horse because their feet are better adapted for traveling on sand or rocky ground (no hooves to pick up stones, split, crack or sink into soft sand), their humps and thick fur are combination food stores, metabolic water store and insulation against the sun's heat, which horses lack. Horses also have higher water requirements than camels.