Okay, here is what you need to beat the E4:
Flamethrower/Fire Blast/Cross Chop
Thunderbolt/Thunder
Psychic/Shadow Ball/Earthquake
Ice Beam/Dragon Claw/Outrage
Lorelei's Poke's go down easily with Fire, Electric, or Fighting moves. Bruno's Pokes are very succeptible to Ground moves, but you can also throw in a Psychic move on the non-Steel variety. For Agatha, only Misdreavus isn't part Poison-type. The rest are begging for a Psychic attack. Use SB or EQ on missy. As for Lance, the Dragonites are doubley weak to IB, Gyarados doubley weak to TB, and Kingdra to DC or even an EQ or two. Watch out for Aerodactyl. It is quick, but FT/FB works well. Use a Water move if you brought one. Lastly is Blue....errrrrr Gary. Definitely easy, but Tyranitar is a *****. DON'T get suckered into using a Water move. Use EQ instead. Burn Heracross and Exeggutor, zap Gyarados and Charizard (Water Char if needed), and hammer Alakazam with SB or EQ. Victory is yours.
BTW - Make sure your Pokes are at least 45. 50 is preferable.
edit: Okay, after looking at your team, I see your problem. Articuno, Plusle, and Loudred are not good choices. Make sure your Charizard knows both Flamethrower/Fire Blast and Earthquake. Bring either Lapras or Starmie/Seadra/Gyarados that knows Ice Beam. I suggest Starmie since it also knows Psychic, is fast, and can handle the Fighting/Poison Pokes VERY easily. Electabuzz, Zapdos (reconsider using Legendaries), or even Jolteon are your best Electric-type choices. Hell, I would even teach your Machamp (after evolving Machoke, if possible) Thunderpunch and use it. Another option is to raise an Alakazam with Psychic, Shadow Ball, and two punches.