I am a lawyer, educated in both common law (US) and civil law (continental Europe) jurisdictions, but not a trial lawyer, and matters relating to evidence are very different internationally. So take what I say with a huge lump of salt. BUT:

Civil Law

From my experience, the burden of proof and the parties credibility will decide cases like this. You as a claimant have the burdon of proof initially, but you bring all the evidence you can -- screencaps, correspondence (your take-down letter and possible replies), witnesses that saw the image on the site (give us a link, we all become witnesses then) -- and you make your case. At that point, the burden shifts to the other party to state that it didn't happen the way you say it did. Since you were telling the truth, that would mean lying in court (=prison time). Many people will not outright lie to a judge, and even if they do, they will not be particularly convincing. So if you got your stuff together and did your homework, you have a fair chance of winning. Most likely, the case will settle at some point.

Common Law (US)

You make your case to the judge and back it up with what evidence you have (see above). This, since you are telling the truth, will likely be convincing enough for the judge to order pre-trial discovery. This will force the other side to hand you over their records, including possible backups of the site that contain evidence of what you say. If they claim there are no such records or tamper with them (i.e. lie to the judge), that is contempt of court (=prison time) and if their attorneys are complicit in it, they are likely to lose their license. So if you got your stuff together and did your homework, you have a fair chance of winning. Most likely, the case will settle before discovery if it is ordered (because discovery is expensive and annoying).