As a former and now retired manager of a global brand of car wax, and an owner of a white carbon bike, I think you have received good advice from many of the former posts. The order I would try is a gentle cleaner, like Windex and/or Dawn. You could move to more aggresive cleaners before the next step. The type of dirt would suggest which one. If you do not know, which is likely, do Windex first. Let it soak for a at least a minute or more no matter the cleaner.

If that fails, try a car polish. Car polish has a solvent and an abrasive. Abrasives can come in different shapes. Some are jagged and some rounded. One would want to start with a rounded abrasive but you will likely not know which product has it. Perhaps you could guess from the claims such as gentler on finishes and such.

Finally, if you are still not happy, you may need to move to rubbing compound. This has a higher concentration of abrasives and solvents (I believe, we did not sell them). . So, it will do a better job but it will also have the highest likelihood of paint damage. Always be gentle no matter your approach.