Now I have been learning chemistry for five years. I remember when I started organic chemistry, it was fun to draw arrows between molecules to show, as if in a mathematical demonstration, how the reactions occurred. In every lesson I had, teachers explained to us how a specific reaction (for example the Shapiro reaction) occurs step by step, explaining the chemistry of each group in each intermediate as if things were obvious (you know how teachers are). 



But I've been wondering for some weeks now how does a mechanism come to be considered as accepted or still discussed? 

If they use some, what kind of spectrometry techniques are used to measure the amount of each intermediate? If not how do they proceed? Do they use computational chemistry? Because for example for a reaction such like a $\mathrm{S_N2}$ it doesn't look too tricky to find how it works, whereas for Fries rearrangement (I don't know if the mechanism is considered as accepted or not) it seems to be more tricky. 



(Ref.)



So can you explain the methods (at least the most used) to confirm a mechanism? I am aware that "confirm" does not mean that we are 100% sure, but rather that it is simply the best we have found so far.

Great question!
When I was teaching, Anslyn and Dougherty was a decent text for this. Here are some general comments:

First, please note that you cannot be sure about a mechanism. That's the real killer. You can devise experiments that are consistent with the mechanism but because you cannot devise and run all possible experiments, you can never be sure that your mechanism is correct.

It only takes one good experiment to refute a mechanism. If it's inconsistent with your proposed mechanism, and you're unable to reconcile the differences, then your mechanism is wrong (or incomplete at best).

Writing mechanisms for new reactions is hard. Good thing we have a whole slew of existing reactions that people already have established (highly probable, but not 100% guaranteed) mechanisms for.

Computational chemistry is pretty awesome now and provides some really good insights into how a specific reaction takes place. It doesn't always capture all relevant factors so you need to be careful. Like any tool, it can be used incorrectly.



The types of reactions you run really depend heavily on the kind of reaction you're studying. Here are some typical ones:

Labeling -- very good for complex rearrangements
Kinetics (including kinetic isotope effects) -- good for figuring out rate-determining steps
Stereochemistry -- Good for figuring out if steps are concerted (see this example mechanism I wrote for a different question)
Capturing intermediates -- This can be pretty useful but some species that you capture aren't involved in the reaction, so be careful.
Substitution effects and LFER studies -- Great for determining if charge build-up is accounted for in your mechanism


For named reactions, the Kurti-Czako book generally has seminal references if you want to actually dig through the literature for experiments.
For your specific reaction, what do we think the rate-determining step is? Probably addition into the acylium? You could try to capture the acylium intermediate.
You could run the reaction with reactants that have two labelled oxygens and reactants that have no labelled oxygens. Do they mix? If not, it's fully intramolecular. Otherwise, there's an intermolecular component and the mechanism as written is incomplete.
A quick Google search suggests that the boron trichloride mediated version has been studied via proton, deuterium, and boron NMR. I didn't follow up on this, but there's clearly some depth here.
When I was T.A.ing for Greg Fu, he really liked to use an example with the von Richter reaction. I might be able to find those references...
