Anyone know anything about paint?

[LargeProsecco]

I have these awful 70's varnished wooden doors, which I'm trying to paint white (pic attached).

So far I have sanded down using 80 grade sandpaper on a mouse sander, then used sugar soap, before using this primer (which. Johnstone's recommended) 2nd pic

However the colour of the underlying paint seems to be soaking through the first layer of the primer (3rd pic?

Should I persevere & do another layer of primer? Or a different primer?

There isn't underlying paint is there, it's just wood and varnish? I'd just put another layer of primer on it when the first layer is dry. Some of it is probably sliding around a bit as it's hard to paint on top of varnish, even when it's sanded.