I need help with Jax.

I started this game sometime around 2012? and started as Jax. I played every game with him back then, and would even insta-lock for him. (Beginner mistakes)

Now I play him, and its like I'm completely new to him. Could I get some tips/pointers on Jax's early-game?

My biggest issue is early-game Jax against other melee champs. I feel like I'm under-powered compared to my melee enemy who beats me under turret. Then they dive the hell out of me till they claim a kill with me not being able to do much. I got rid of the Conquer rune as it literally made things even harder for me for some reason, and went back to the first rune in the row of runes Conquer was listed in. Late game isn't a problem (if late game means playing till the 25-30 minute mark) for me, but I love flourishing early-game. I'd hate to be Jungle reliant every game. I've tried building Hextech first, Bork, and even going tanky first (A Sion was running my ass over with a real weapon, and denying me farm under my own turret).

Not complaining, just wanting some input, because many people are saying Jax's current state is fine, and if not slightly over-powered, but I feel so hopeless, and weak when I engage 1v1 early making sure to utilize my stun, and everything. I ward hop, go in with stun if the situation calls for it, wait an enemies attacks out for the right time to proc stun, etc.

So far the best safest way to do anything is to bait them to try to kill me under my turret when I'm low hp, and stun them, but that's ridiculous.