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.
She has remained loyal to the White Tower during the split.
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.