The system is not unfair, it's that you aren't playing well enough.
Alistar is a decent support, but lacks what you need.
Alistar lacks the ability to carry and make plays on his own. He can do 1 stun combo that needs follow up, then what? If you get no follow up you are done, there's nothing Alistar can do as a solo player.
What you want are the likes of Sona, Nami, Soraka, Lulu, Janna (she's freelo right now) or Brand, Veigar, Lux, Malzahar, Malzahar who are quite OP at bronze.
The reason for the first 5 is that they will just keep your team alive. Since it's bronze, you don't use your ultis to make engages as Sona or Nami, you use them as follow up after a team starts, and both Nami and Sona pack the damage to almost solo carry, while a decent Janna is like "were you trying to kill my team or something?"
As for the last 5, they do the exact opposite, they don't keep your team alive, they keep the enemy team respawning. You are just so aggressive and having so much damage that the enemy team can't deal with you if you position right, meaning, at the back of your team, and since your CC is so insane as well, with slows, stuns, roots, the enemy team can't really focus you either. All you need is to play it correctly.
That's what you need on supports.
Not supports who need a duo partner to funtion, but supports who can force the team to be the duo even if they don't want to (Janna, Nami, Sona, Soraka) because you just don't let them die if you play it decently, or you just deal so much damage that the enemy team can't respons (Brand, Veigar, Malzahar, Lux, Morgana).
The fault is on you for not having the right thing for the right situation. And by situation I don't mean match-ups, but division. Alistar is a good support, same for Tresh or Braum or Leona, but they just need others to function, which you don't have in bronze. Everyone wants to be the one to carry, so instead, you either make sure they do it with the first set of 5 defensive supports and just make sure your team ain't dying, afterall, all they know is to deal damage, so the one who wins is the one who deals damage in a longer period of time, so if you make sure that your team is the one surviving more than the enemy team, you basically win more, while the other set of offensive supports is just making sure that the enemy team is having less time to deal damage because you deal way more damage.