How do I dispute a traffic accident police report, as damage to my vehicle is inconsistent with the report?

If your insurer is not paying your claim (e.g. under collision insurance, assuming you have such) despite you providing evidence (possible evidence discussed below), you would sue them for "breach of contract": for not honoring their contractual (an insurance policy is a contract) obligation to pay for your loss. 
If the other driver's insure will not offer you compensation, your recourse is to sue the other driver (you sue the driver, not the insurer, because their insurer is *their* insurer--it has an obligation to defend and/or pay for them, but does not have any duty or obligation directly to you).
If an insurer or the other driver wants money from you, you can refuse to pay--they'd have then the right to sue you, and you'd have the right to defend against their case in court.
So in one way or another, if you don't get the money you want, or someone else wants money from you, this may well end up in court: the only way to force anyone to pay with by a court order or judgment, and until/unless there is a court order or judgment, any payments by anyone are voluntary.
In court, both sides present testimony from witnesses (including the drivers), possibly from expert witnesses (e.g. any insurance adjuster or accident reconstruction experts who examined the car and the other facts), and also other documentary evidence, including police reports and photographs. The police report is NOT a legally binding determination of fault: it is the police's opinion as to what happened, but does not have any legal power itself. It is evidence only, though it can be strong evidence, since the police are generally trusted in this regard.
Therefore, to fight the report--in court, if it comes to that, or just in negotiations, to try to convince an insurer to pay voluntarily--you need strong evidence, too. You start with your own testimony, but it won't be enough? Were there other witnesses, whether passengers, pedestrians, etc. who will support your version? Get their testimony, too. Take photos of the damage, the intersection, etc.--anything that supports your version. And you may need to hire a private insurance adjuster or accident reconstruction expert to render an opinion. Of course, hiring such professionals costs money, possibly a lot, so depending on how much is at stake, it may or may not be economically worth doing this.