Citizens do not use the road I made for them in Cities Skylines

1. The road appears to be much longer than the alternative. The wide curve towards east does not seem to be necessary. It should be possible to build it much shorter.

2. The road does not seem to solve the actual traffic problem. Most of the cars passing through the heavily congested area seem to want to go south, and those which want to go north seem to want to change from the westmost road to the eastmost road. The road with the dark blue arrow doesn't seem to have any congestion, so the light blue one isn't really needed as an alternative.

What I would recommend in this case is to feed all the four roads coming from or heading to the north into a roundabout. There already appears to be one in the south. And then I would create a bridge to accomodate specifically those cars which come from the north via the western road and want to go north via the eastern road. But there doesn't seem to be a reason to do this here. It should be possible to do this much further north. Perhaps over the cloverleaf exchange in the north, or even further north.

It seems to me that your biggest problem is that the cars merging in from the two left-hand roads coming from the top of the screen don't have enough time to choose lanes after the two roads merge and before the intersection where they need to either turn right into the roundabout or go straight ahead back towards the top of the screen. Just moving those two intersections further apart might help.

But I also agree with Philipp that the whole thing looks way overcomplicated. This is how I'd redesign it, at least to start with:

Now the short two-way road segment coming off the roundabout is gone entirely, along with the problematic intersection at its end. The two roads coming from the top of the screen now merge long before entering the roundabout, giving traffic plenty of time to merge and choose lanes, and lane math should guarantee no traffic jams at the merging point.

Note that any traffic that wants to come in from the top of the screen and go back up now has to go through the roundabout. But you shouldn't be having much traffic taking such a weird route anyway — or, if you do, there's likely a problem somewhere else that you need to fix.

(I have noticed that the game sometimes generates weird external traffic that just wants to make a U-turn and exit the map the same way it came in. If that's what's happening, you could probably handle it just by adding single highway ramp from the incoming side to the outgoing side of the highway near the map edge. No need to let that useless traffic get anywhere near your actual city!)

Ps. If making the middle roads single-lane creates problems further up at the cloverleaf interchange, I'd also consider keeping them two-lane up to the point where the first ramps of the cloverleaf branch off, and maybe even all the way across the cloverleaf. But if I was designing this from scratch, I'd definitely first try using one lane only for them. Or even redesigning the whole interchange at the top of the screen so that the roads going across the left-to-right highway are kept entirely separate from the on- and off-ramps to that highway.

(Right now, you only have partial separation: while vehicles going across are presumably supposed to use the bypass bridges over the cloverleaf, they can instead take the middle roads through the cloverleaf and create needless congestion there.)

Frequently, in Cities: Skylines, having more lanes than you need just pushes problems to where those lanes end. Instead, a good set of rules of thumb is to:

1. use no more lanes than you need,
2. keep the number of incoming and outgoing lanes the same at least in busy intersections, and
3. keep intersections on multi-lane roads as far apart as possible, so that vehicles have time to change lanes (and also so that, if a temporary traffic jam does form, it doesn't spread across multiple junctions).

FWIW, the cloverleaf interchange fails the last rule, which is one reason why it's not so great for high traffic volumes, both in-game and in real life.