Irelia vs Jayce help and tips?

Ok, so ive been playing alot of Irelia in Normals lately and today i ran into a Jayce top. From what i found Irelia "counters" Jayce but i couldnt figure out how. At first i started with 2pots and a flask and managed to "outplay him" for the first 2 kills by using the brush and constantly moving around him to avoid his Qs damage as i noticed from the start that it did a little under half my hp when he managed to hit me. He would harass me with autos constantly as well whenever i came in range. I tried punishing him when he missed but half the time i Q - E'ed him he'd just knock me back and kite me and win the trade. I will admit i was frustrated. Should i have not all inned him after he missed his QE at full hp? Anyways we ended up going even until after about 2 kills when he disengaged while we were both low and landed a lucky q. After that i roamed bot and got a kill but he followed me there with Tele killed me and ended up getting a triple after my jg tried all inning him while he was low making him 6/2. One thing to mention is that niether of our junglers ganked top at any point in the game. Anyways is this lane winnable as an Irelia player? Can i get some tips on what i should do in this matchup and how i should play it? Was rushing phage a bad idea? What should i build instead? How can i outplay him? Thank you in advance.