Turanna is not particularly strong in the One Power. Her level of strength is described by "The Wheel of Time Companion" as 26(14) which is a middle level in Aes Sedai hierarchy. With this level of strength she is unable to open alone a suitable gateway to Travel.
Turanna was one of the captured Aes Sedai to be interrogated by Verin Mathwin in order to discover the White Tower's intentions towards Rand al'Thor. In reality Verin used her mild Compulsion to implant in their minds some instructions about Rand.
Later it is reported by Sorilea that she along with the rest of the Aes Sedai prisoners have sworn fealty to Rand.
As with many Aes Sedai sworn to Rand, Turanna probably fought the Last Battle on the slopes of Thakan'dar. It is not stated whether she survived or not.