It appears that your therapist is willing to take your suicidal thoughts seriously, which is good. However, it is important for you to be able to discuss your suicidal feelings and intentions comfortably, objectively, and in detail with your therapist.
Usually experienced psychiatrists and professional psychotherapists have worked through their own fear of death in the course of their own psychotherapy or analysis. Some less experienced doctors and psychotherapists are still terrified that they might be responsible for the death of one of their patients if they intervene. This fear makes it difficult for them to help you, especially in a crisis. If your doctor or psychotherapist is not comfortable discussing your suicidal thoughts and emotions, you may have to begin looking for someone who is.