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.
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.
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.