Internal linking is important for SEO, but as my website grows, it'll be impossible to comb through all the internal links.
What are some tools/software analyze my internal linking structure and maybe even provide a visualization of it?
There are a couple that come to mind. Screaming Frog is pretty well known in the SEO of its crawling capabilities. You can get some value out of the free version but to do the visualizations you might need the paid version.
I have also used PowerMapper in the past. This was great for visualizations as they have a lot of different outputs. This is also a paid tool but worth it in my opinion.