Hi Mustard bite,
The /z/ was a very difficult sound for me too. I cannot tell how many hours I spent listening and repeating dialogues until the teacher accepted my pronunciation.
Here are some tips and part of a dialogue from the book “How Now, Brown Cow?” by Mimi Ponsonby.
/s/ This is a voiceless sound. Place the tip of your tongue between your teeth so that the teeth grip the sides of the tongue firmly. Now draw back the very tip and press it against the bottom teeth. There should now be a small passage over the top of the tongue through which air can pass straight from the lungs. Keep the lips spread and expel the air in a hissing sound.
/z/ This, like its voiceless equivalent /s/, is a continuous sound. The lips and tongue are in the same position as for /s/ but the vocal cords are vibrated, which will cause some tension in the tongue itself. The vibration should be very strongly felt.
DIALOGUE 15. The zoology exam´s on Thursday
EZRA: How´s things these days, Lizzie?
LIZZIE: I’m exhausted. Revising for the zoology exam!
EZRA: You´ve got bags under your eyes, Lizzie. Take it easy!
LIZZIE: It´s all very well for you to advise, Ezra, but I’m going crazy. One of those miserable Zeno boys, two houses down, plays his transistor as if he was as far away as Mars!
EZRA: Boys will be boys. These days everyone plays transistors…”
Some useful links:
YouTube - Accent Reduction: English Pronunciation of S and Z
BBC Learning English | Pronunciation Tips