The sense of self or the "I" is part of that stream of thoughts, that is why you can't stop thoughts except by fully concentrating in something; the intention of stoping thoughts and the self that wants to achieve something are thoughts themselves. An approach that I have found very powerful is to direct atention to nervous system activity which is an objective form of conciousness, something like the sense of touch but extended to the inside of the body (it is called kayotsarga in jainism), besides gaining the ability to fully relax at will all parts of the body it brings awareness to a whole new level.