Some students hate to use a textbook because it brings back memories of high school English. In Japan, where I am, that is often the case -- and understandably so, when high school English was heavy on grammar, reading, and writing for the uni entrance exam. The best thing about a textbook is the structure it lends to a course. But there are ways around the "no textbook" request.
As others have said, assign a topic each week. I usually give an article as homework from my website. The student reads it, understands it, and comes to the next lesson ready to talk about it. For some of my students, I focus on vocabulary, which means one or two pre-lesson assignments. For others, I focus more on fluency, so we then do a lot more with speaking activities. In either case, the lesson is structured, meaning I'm always working towards an objective in mind. An article of 300 words often helps the student along, because it serves as an introduction to the topic. A series of questions on a single topic also achieves this.
Compare this to a free conversation. Without having thought about a topic and/or prepared for it, how can an ESL student challenge himself with new vocabulary (particularly higher-level words on a specific topic)? How can he form intelligent responses once the discussion moves to concepts he hasn't necessarily thought much about in his native tongue? Instead, the lesson usually hovers around familiar ideas, grammar structures, and vocabulary. Time also gets wasted as the student frequently checks a dictionary.
Pre-assign a topic; give a brief article and/or questions; work on specific activities with an end objective in mind: these should give the lesson enough structure to assess, challenge, and improve your student's abilities, yet still skirt around her apprehension of a textbook.
Chris Cotter www.headsupenglish.com