Matchmaking places you in games where your MMR is similar to the average MMR of all players. So if your MMR is, for example, 1200, and everyone is solo, you can reasonably assume that everyone is somewhere between 1150 and 1250. In fact the range is probably tighter than that.
However, in normals you can queue with anyone. That means a Bronze player (MMR: 800) can queue with his Diamond friend (MMR: 2000) and they'll both get placed in a game with an average MMR of 1400 (Silver/Gold).
How much MMR you win/lose is proportional to how close you are to the average MMR of the game. So in our first situation, you'll probably gain as much MMR for a win as you'd lose for a loss. In the second situation, the Bronze player would gain more MMR for a win and lose less for a loss, while the Diamond player would win less for a win and lose more for a loss.
In your scenario, if your MMR was higher than your friend's, then you probably tanked a lot of MMR.
HOWEVER, and this a big however, with the exception obviously of premade groups with disparate MMRs, if your MMR is less than what is an accurate representation of your skill, your games should be easier not harder. Whether you're more mechanically skilled or you have a better understanding of the game and how to close out a win, you should have the advantage. If you don't feel this advantage - if it feels like you're unable to affect the outcome of games by a meaningful amount, that is an indication that you are actually correctly rated by the system.
Also, judging from your match history, ignoring the AD Soraka game, your last hitting alone is probably Iron or Bronze level. sub-100 CS in 20+ minute games is god awful. I think you significantly overestimate your own skill and have a propensity to blame your own inadequacies on your team. I'd recommend fixing that, as you can't improve if you think you're not at fault for your losses.
