Shouldn't sitemaps make indexing my site quicker?

Hi,
So Google is indexing, unindexing and reindexing the web pages of my website like is expected for a new website, but I'm surprised it hasn't yet indexed my deeper pages (which is a huge majority of my pages) despite me uploading a sitemap ages ago. Google has now indexed only those sites 2 links away from my homepage, when most of my pages are three links away. Is sitemaps doing its job? is there a problem with my sitemap XML? I submitted the sitemap around the 2nd of this month and my site got first indexed around the 6th or something like that.

Grand.

 

Check the status of your site map, and see what is the number of URL submitted, this might give you an idea how many pages are submitted via sitemap and if they are fewer than what it should be, some pages would be missing.

Revise your sitemap and make correction if needed.

 

Sitemap shall only direct spiders to crawl your pages, but crawl doesn't mean indexing.

 

Doing a sitemap doesn't guarantee indexing. The best way to achieve having your site indexed is to have other sites link to your site.