Biggest problem is the grey primer. You really need a white primer for the best results. Once you have that, then apply the white in very light coats and allow plenty of drying time between coats. It's the only way with Tamiya TS sprays as they are very, very thin. Build it up very slowly or this will happen every time..
I would consider a cheap basic airbrush if possible. You could use the Zero basecoat Pure White. It will cover very well, even over grey primer in just a few layers. It makes the job so much easier. Tamiya TS paints are very translucent. They will take on the colour of the underlying plastic or primer. It can even change the colour. The reds are really a good example of this, needing a nice, white undercoat to achieve their best and true colour.