I have a new domain and I am beating pages with much higher page ranks / DA etc
Typically in the niche I am targeting if I use the Keyword Difficulty tool i get an average of about 50% - I think this tool mainly uses DA / PA to work out the difficulty %.
After I will do a Google search and look on the pages for mentions of the keyword's I am going to target with my pages / posts.
Often I find there are few mentions or 1 exact match mention in the content with the page title being something different / not exact match.
I will then build a page which is targeted specifically for the keyword and optimise for it, I don't over do the optimisation - if the other pages only have 2 mentions of the keyword in the content I would normally build a post with say 3 - 5 mentions. I have noticed when going over 5 keyword (aprox) pages tend to rank poorly or rank badly then crawl back up the SERP's slower - this could be due to the domain on the site being 1 month old.
I also only build quality content that is relevant to the search term, this should prevent the pages dropping from the SERP's (I hope!).
Obviously if your niche has highly optimised pages and a bunch of links pointing at each page then this method is not going to work.
Hope that helps.