How to make a turning lane without an intersection?

With the two way roads, the game's behavior is always to create an intersection like you're seeing, as it assumes that you want traffic from both directions to be able to enter/leave the intersecting road.

So instead of using a single two way road, you can get fairly close to what you're looking to make by using two one way roads as a sort of DIY avenue. It ends up looking something like this:

At this point, you could add your one way 'turning lane' to the one way road that you want it for. It would still create an intersection, but since both roads are one way it would work essentially like what you're looking for, with cars peeling off out of the right lane to avoid the roundabout. You can make sure this happens by using a highway on-ramp for the turn lane: cars try to choose the fastest routes and on-ramps have higher speeds than regular roads.

Edit: here's one that I built:

Note that I was using the Sharp Junction Angles mod.