The existing mesh works, but its a mess, not a mesh ;-)
Version 1: Low poly.
This is topology which is focused on small amount of polygons. Edge Flows aren't so good and using Proximity Loops will be pretty hard, but nevertheless both methods are working best with Bevel Modifier (Angle) + Subdivision Surface.
Version 2: Edge Flow.
This method is focused on proper Edge Flows. You can use here Proximity Loops to sharpen edges but as stated in Version 1 it's better to use modifiers.
Here we need to use two ngons to stop inner loops before 'big circle'.
How to make it.
I will not create whole step-by-step answer here as methods used was just Blender 101 modeling. I've started with outer Circle then created another one in the middle, scale it to have egg shape and from there it was just manual Extruding, Moving, Joining etc. I've also used two add-ons here: LoopTools > Space and F2 to speed up my work. They both are built in in Blender, you will just need to turn them on.
My model is pretty uneven as I was modeling it 'by hand' with distorted reference image. If you have good .svg shape, you can import it and use it edges to snap vertices to them.
Here is my blend file. You can check different layers for both versions, also you will be able to see detailed values for modifier I've used.