I'm a Jax OTP - and I wouldn't even say Malphite is his biggest counter anymore. Malphite is his classic counter - and has been since like season 2. However, with conqueror now, and the free access to tenacity (the legend rune), Jax can get 20% true damage and 50% tenacity, which really brings this matchup closer than it used to be.
The inital laning goes 1 of two ways, depending on the Malphite player - he will either run a mana crystal or flask and look to Q poke (with comet). Jax needs to sustain best he can and accept he will lose early trades. But malphite will give up some tankiness to do this and will be easier to fight in the mid game.
OR, malphite goes Grasp and looks to be a tank (usually starting shield), and in this situation Jax can look for early trades (trying to bait out malphites E) - but mid-game will be rough if Jax can't get a small lead or go even.
Either way, Jax has a chance now. Whereas, for example, in season 4 or 5, this was so one-sided for Malphite it wasn't even funny. It still is a good matchup for Malphite but just not as good as it used to be. These are good against Jax too:
If someone takes my Jax, I take teemo. So, so easy with teemo. Bully, harass, push under tower, pressure, deny cs and kill. Sit on shrooms so he can't jump on you and put a couple of levels into blind. Easy. Hardest counter IMO.
(Tricky one - but IF nasus gets a lead, he can absolutely destroy Jax with his stupidly buffed ult Q)