you don't always need to rush those 2 items... jax can be played different depending on the team comps.
as odd as it sounds ive found vs heavy harass ADs top that iceborn is a better choice than trinity mainly because you lose a bit of burst, but you gain so much more sustain, and have literally infinite chase with the slowing pool.
I prefer to get the life leech / slow part of the blade, then rush trinity or iceborn, then go for a giants belt then finish out the gunblade.
jax without HP is laughable at best. you need that giants belt before you finish your 2nd item regardless of what route you choose