Oklahoma had a decent run from 1905 through 1926 under Bennie Owen, who won 122 games. But he lost 54 and tied 16, so his winning percentage is only .677. And, other than that, there wasn't much to cheer about.
Over your time period, Oklahoma went 9-21- vs. Texas. And 8 of those 9 wins came under Owen, who went 8-8 vs. the Horns.
There were a couple of claimed Southwest Conference championships in there. I say claimed because there was not a set number of conference games. OU claims the 1918 SWC championship with a conference record of 2-0.