How can I improve my mobile menu layout for better conversions?

Hi all. I am hoping to have a custom solution to my mobile problem.

I have ALOT of product types and collections, and while my desktop mega menu is okay - the mobile version is not optimized and I feel is not generating alot of conversions. See HERE

I was hoping to have it changed to look and function similiarly to this one HERE (without any apps)

Can someone please assist or offer advice on what steps I should take?

Thanks so much