Don't go in on Heimer while his turrets are up. It is what he wants you to do so he can hourglass or R+Q.
Fight his turrets with help from your minions. Prioritize magic resist, a
will give you good defense against him and give you more mana regen so you can poke his turrets out of commission. Contrary to what you might want to do, play hyper-aggressively from the start so he can't get all 3 turrets up. Letting him get all 3 turrets up early game will let him harvest insane CS and make him almost impossible to fight.
Worth a shot: taking
midlane is an underestimated strategy vs. heimer. He regenerates a turret every 20 seconds, and you will have smite available every 30 seconds. It will make your life easier, and will also allow you to get a high gold efficiency item and also a nice CC if you take a blue smite. And if he has zoned you to the point that you can no longer CS, you will still be able to grab most of the cannon minions.