Calgary used to probably be the most conservative large city in the world. It’s still pretty conservative as far as large cities go, but it’s gotten more liberal (and diverse) over the last decade or so. In the U.S., I would probably rate Oklahoma City, Colorado Springs, or Mesa, AZ as the most conservative large city.
Since it adopted "metro" government--merging the city government with the surrounding county--in 1970, Indianapolis has had mostly Republican mayors although the Democrats took complete control in 2015.