You have a problem with people turning left at that first intersection. If everyone turns the same way, it effectively turns the road into a single lane road. Address this by making sure that people who turn right and go straight can still reach destinations on the left. Cars do not always take the shortest route, so some will take the alternate routes.
Do you see that square at the entrance? That's a conversion intersection between a heavy avenue and a streetcar avenue. Making a streetcar avenue at that point is useless - people aren't just going to jump out of their cars and ride it. Since you can't upgrade the entrance road to streetcar avenue, downgrade the road between the conversion intersection and your first turning intersection to heavy avenue.
Add destinations that incoming sims will want before the first turning intersection.
I landed in a similar situation in my game today. Here's a shot of the commuters. It's a nice mix of workers and shoppers which have founded a bumper-to-bumper parking lot outside my city. 9pm? That's not a rush hour, it's a way of life.
![entrance traffic woes](https://i.stack.imgur.com/LGhU1.jpg)
Everyone is turning right at the first intersection. I tried various things to solve it. Reducing the number of commuters. Adding a commuter train. Adding 48 commuter buses (at ~5000 simolean monthly cost). Nothing stuck, so I removed those buses and made the roads "worse". Here's a shot of the solution, along with a nice long column of flowing traffic. It's 8 o'clock and I don't know where the traffic went.
![entrance traffic solved](https://i.stack.imgur.com/0U2IZ.jpg)
The fix addressed two issues.
Cars were turning right at that first intersection and immediately getting hit with another stoplight. Throughput was limited by the synchronization of those stoplights. In my fix, there are a few 2-way intersections after that right turn. 2-way intersections (and conversion intersections) have no stop mechanism: cars just flow through. That driver may turn right, but it will be a long time before he hits another stopping mechanism.
But that's not enough. By itself this would just move the problem to the end of that long street. I moved the right turn forward, and made cars have to drive back to reach places near the entrance. Cars also would enjoy a shorter trip to places in the back of the city if they just would go straight through that first intersection. This had the affect of dividing the traffic. In that first intersection, get 3/4 of the traffic to go straight (using three lanes!) and 1/4 to turn (using one lane).
Breaking the grid this way made the drivers do what I want, even though they could have done it themselves. Now, they have a longer commute, but less time spend parking during it.
Best Answer
It really depends whether or not you have rush hour installed. I do, however I played without it for a long time before that. If you don't have it, make a design in which local roads or streets are in a design where it is either impossible or inconvenient for them to be used as thoroughfares, which can help reroute traffic onto thoroughfares.
In rush hour it is a lot easier. You can use one way roads to make a pattern where it easily enables traffic to get in and out of an area quickly and onto the main roads, but do not enable traffic to cross the area easily unless they are on a main road. Also in rush hour public transport helps a lot more than it does in the vanilla version, due to some changes to do with they way sims commute. If this doesn't work, just use streets, they are slow and so for the most part sims will avoid them. Note that this doesn't work in vanilla SimCity 4.
In both cases, the thoroughfares could be lined with commercial zoning, as traffic increases desirability for both commercial services and commercial office buildings.