You have been experiencing a lot of things that I have noticed about my own game. I don't play aggressively enough to be a true assassin/damage bruiser type like a Katarina or a Tryndamere (both champions that people have said are very easy to use to climb out of bronze). My playstyle is all about farming and picking my fights to put me into 2v1's and other skirmishes that I have a clear advantage of. I focus more on building the vision advantage and keeping my eye on the enemy team through wards. If I don't see someone, I assume I am about to get ganked by a full 5-man team. That being said, there are some solid champions that I have found to be very good staying safe in the early game and then having a strong impact in team fights late game. These champions have great abilities to keep you alive and safe:
Orianna - This champion (on sale at the moment of me typing this) is a very good champion in the mid lane. With some practice with her ball, you wield a very powerful tool throughout the whole game. You can do damage, you can shield yourself/teammates, you can speed up your movement speed, you can slow your opponents, and your ultimate has the opportunity to really change the game when you lock up two or more enemies with it. Mechanically, she is easier to play than say an Azir or Leblanc, and item wise she is very easy to build. Start with your Athene's Unholy Grail, then move into a Deathcap (if you're ahead) or a Zhonya's Hourglass (to further keep yourself safe in those team fights). She is one of my favorite picks when I don't know who I am laning against because she can be a safe champion.
Morgana - This champion is capable of playing in 3 of the 5 roles and can really help you and your team throughout the entire game without falling off like some of the other champions. She can shield teammates and make them immune to spell effects (like hooks and stuns), she can do excellent AoE damage and push waves of minions, and she has the ability to lock down a target for a very long time with her Dark Binding. She used to be a big ban in Bronze last year, but there have been so many other champions that people have been banning out (as well as some of the nerfs to her kit making her a lower priority). She is definitely a strong pick still, though, and if you rush a Zhonya's Hourglass on her after a Morellonomicon, she can really change team fights (since after you proc her ult, she can Zhonya and her ult will still go off).
Malphite - This is my jungle main. He is simple mechanically and is very easy to pick up and play. His passive is a shield that scales with 10% of his max Health, a stat that he already builds. It refreshes after 10 seconds if he takes no damage, a timer that actually decreases as you level down to every 6 seconds it refreshes. This keeps him strong in the jungle, even though he does not have a lot of gank potential until level 6 when he gets his ultimate. After rushing Cinderhulk and Righteous Glory, though, you will have a huge shield and be able to start building armor and magic resist without even taking a non-tank item. Late game, your role is just to soak up damage, romp around in the middle of the enemy team while using your Ground Slam, and then focusing your ultimate on getting as many carries as you can in it. Many times, I don't use my ultimate to initiate the fight. Instead, I use it to keep the carries from backing out of our initiate and knock them up for my own carries to eat alive. Malphite is capable as a support pick, as well (although I do favor ranged supports to melee ones), but he is also a good Top laner. A Malphite teleporting into a bush and hitting his ultimate on the bot lane can be extremely disruptive.
Lulu - Yeah, I said it. This champion is not a carry champion, but if you play her in the mid lane or the top lane, she can build full AP and her shields and well-timed ultimate can really turn fights around. She isn't even a late game scaling champion. She has full utility as early as level 3: a skill shot slow with her Glitterlance, a movement speed increasing Whimsy (that also acts as a disable polymorph if used on the enemy instead), and a Help, Pix! shield that can deter some damage or be used as a damaging skill on your enemy instead. Then once you get your ultimate, you can buff up any teammate to give them bonus health, knock up any enemy near them, and give them a slowing aura that lasts for a few seconds. Her kit plays very well into keeping people alive and safe, playing passively to farm up, and play for the team fights. She is a very valuable support pick, as well, since she doesn't need a lot of big items to be a valuable asset to the team. I played her more last season than this, but I have really found great success with this champion when I know that there are other members on the team that can build full damage while I help keep them alive.
I hope this helps you out a little bit. Obviously, this isn't an immediate solution, as you have some more champions to practice. However, if these are a part of your champion pool, I think you will have a good chance of making the kind of progress you are looking for. Good luck on your climb.
