There was a point when I suddenly realized I was improvising a taksim with my dance. I was writing my own melodies.

When responding to the music, we tend to follow either the rhythm or the melody. Often, we follow both.
I often see dancers attempting to respond with some movement to every little drum beat. Sometimes, these dancers find themselves at a loss when presented with music that is more melodic than rhythmic.

Try this... Put on a track that features only drum. Start writing your own taksim or melody with your movements.
Be the melody over the percussive background.

Try also working in a more abstract way with beginning taksims (the taksim at the beginning of a tune that has no drums behind it.) These are more arrhythmic and abstract. Their meanderings cannot be predicted. Try expressing the feeling of the taksim rather than attempting to "keep up with it".

Study taksim. On Turku's first 2 CD's, listen to the violin and the tar as they engage in conversation.