You are playing against your skill level. MMR in norms is meant to produce a 50 /50 chance of winning by grouping players of similar skill together. In ranked you are stratified by the ranking system and can climb as the rank overrides the MMR function of norms. If ranked were matched like norms then you would find it almost impossible to climb as after about 15 games you would be locked into a 50/50 ratio. That would be really bad for the game as part of the experience is the ability to climb in ranked, and people hard locked would not stay with it very long. You could "win" more and improve your MMR, but in norms its really difficult because of the stacking of teams to produce the 50/50 win chance. Ranked matches have no bearing on norms just as norms don't affect ranked matches.
The best thing you can do is premake with better players to increase the MMR of a match and play up to that level. Playing meta champs and builds to increase your chances for a win also helps.
As far as those guys in bronze and silver in your games it just means that they are the same MMR/ELO as you in norms. They may not have spent as much time grinding ranked thus the lower rank, but as far as norms go they are your skill level.
This all comes from a service ticket I sent to riot in the past asking essentially the same question.