As other posters note, the plot of Bagdad is incoherent, though its momentum and the good-bad IDs are always clear. It's one of those odd paste-jewelry gems of the mid-century Hollywood studios. Overall the prevailing aesthetic is camp, especially given the outrageousness of Maureen O'Hara as a red-haired, green-eyed, candle-cheeked Bedouin princess and the languid Swede Paul Hibschmud / Christian in the Valentino slot.
But if Bagdad is campy junk, it's not exactly cheap junk. The color is touching, as are the efficiently managed sets. If you look closely at the California desert shots, you'll see many a rock outcropping used in Western chases with cowboys.
Bagdad is not a Western, though, but an "Eastern." As another poster noted, Maria Montez was the icon of this minor movement in Hollywood genre films. As with other such potboilers, one of the pleasures is the precise performance of the character actors, e. g. John Sutton as the villain and whoever plays the sentimental role of the old retainer to the princess's father.
Plenty of credit to O'Hara and Price for carrying the film. The scenario has so many lurches and fillers that you see something in these actors beyond mere talent, though both have plenty. Beyond talent, they're both troopers who never flinch when the script double-clutches--they brave every scene through, holding up the pretense, and O'Hara's song-and-dance performances give some scenes a surprising robustness.
Given the plot's weaknesses, I doubt if this movie is worth sitting down for 90 minutes at one stretch. But I taped it off a cable channel and watched about 10 minutes at a time. Virtually every 10-minute episode featured a song or dance, a scene of intrigue, and luminous backdrops with well-staged action. Obviously a viewer must have a predisposition to pure escapism. What else is Hollywood for? Recommended on those terms.