Hey man, I was an s3 who has made it to n2 and I think I've noticed the same thing. Funny thing is I think the disparity between s3 to s4 is bigger than s4 to n1. The biggest difference that I know of is team work. If I remember correctly s4 is the rank of n1 drops and really good s3s, so they have the knowledge of team work from n1 (which makes playing nova really fun btw) and then you have all the s4s stuck from going into n1 that have plenty of head shotting lower silvers.
If you're looking to improve your skill I have 2 suggestions. If it's just about rank than I have 2 other suggestions.
If you are trying to actually get better at the game then practice deathmatch no shot bots, head shots only with m4a1s at least 30 minutes every day. I swear to god, the muscle memory build up for practicing head shots is great. m4a1s has some of the best accuracy for rifles, and it also lets you spray, and it only comes with 20 in a mag so you get to practice mag loading safety, head shot accuracy, and spray down as much as possible. Try to shoot like the guys in c9 do. Go for the headshot and learn to spray control down if you miss the first 2.
Second advice for skill improvement would be to watch smoke videos and try to practice on communication in games. Once you learn the smokes for popular maps, always start the map with "I'm going to save" or "I'm going to buy an ___" and then "I'm going _. I'll smoke _" and then do it. You will be amazed by how many people work around that kind of knowledge. And then when you die make sure to say "he killed me on __ I lit him for _. he has an ___"
Now if rank is all you care about, then here is my advice for that: Whenever you play with people, pick the highest fragger on the team and friend him. If it's you then pick the 2nd highest. If you make 1 friend every other game, then by the time 1 week roles around you got yourself an army. Just make sure to always group full 5 man with those guys and only play with them. Rotate out the lowest guy, or if that's you, the second lowest guy, and just keep on playing. If they rank up, then don't play with them. Rotate in another person. Rinse and repeat and you will rank up pretty quickly.
If that sounds like too much work you can always try playing with smurfs and get farmed up. But I honestly think smurfs are bad for the community so I don't really recommend this option at all. I consider smurfs as bad as cheaters actually: just toxic and poison imo.
Even though I have given you 1.5 suggestions for focusing on rank (first one is valid, second is kind of shitty), I really hope you try to focus on skill instead of rank. It really doesn't matter what rank you are, as long as you enjoy playing the game and getting better. I think a lot of people mistake their wish for wanting to improve and get better with that tiny badge on the right of their name. The badge is just a showcase of your improvement. It is a reflection of progress, not progress itself. The feeling of greatness and success does not come from the image, it comes from the feeling of knowing you've gotten better by practicing.