For example, a thief steals a loaf of bread and the owner of the store can gather a mob to lynch a thief. Anarchy has the great potential to administer unproportional justice.

You shouldn’t come into an anarchist community and start answering questions about anarchism when you clearly haven’t done your homework.

But hey, since OP is interested in how anarchy would work, let’s go over how such a society would respond to the scenario that you’ve painted. Vigilante justice is never impossible in any society, but that doesn’t mean it would be tolerated. The requirement to have disputes arbitrated by a neutral third party is pretty universal. What differentiates anarchy is that arbiters are freely chosen by (possibly delegated) mutual agreement, instead of the state forcibly inserting itself into every dispute as the supreme arbiter.

Let’s say the thief was a member of a commune. Since the thief is dead, their dispute with the baker and the lynch mob can be claimed by their next of kin, or closest equivalent. Either way, we’ll say that the dispute gets delegated to the commune as a whole, which collectively handles security and dispute resolution for its members.

The baker has a contract with a company (probably organized as a workers cooperative) that offers security and dispute resolution services. For simplicity, let’s say that the members of the lynch mob also use this company’s services.

The commune and the company might have different sets of rules that their members agree to, but it’s reasonable to assume that they both recognize:

• The thief should not have stolen the bread, as it was a product of the baker’s labor and was not being offered for free. While the commune functions primarily via gift economy, they defer to local norms in these situations.
• The baker would be entitled to restitution for both the stolen bread and the costs necessary to secure that restitution
• Killing the thief to stop them from stealing in the moment would have been a wildly disproportionate response
• This was not merely done as an act of immediate defense, but an act of retribution
• The baker made no attempt to resolve this dispute through a neutral third party
• The members of the lynch mob all acted as accomplices to the murder

From there it’s just a matter of negotiating what restitution is owed to whom. Perhaps the commune and the company can’t come to an agreement on what exactly is owed, so they agree to defer to a neutral arbiter of their own. They may both be members of a local federation of dispute resolution bodies, which would simplify handling this.