I imagine there will be some debate here about exactly what goes where, but I feel with the evidence I've collected thus far, it'll be pretty hard for some to deny that the broadcast order can't be the true order of events. I certainly don't claim I know the true order (basically impossible), but I make a lot of educated guesses about certain episodes and general placement in relation to others.
This is an important start to this claim. If we have no idea how long it's been between the first episode and the last episode, we'd have no idea how to compare how long it's been between individual episodes. Luckily, we have one important annual event that's happened twice throughout the series: The Summer Sun Celebration. In season one, we start off the series with Twilight planning for Celestia in Ponyville, heralding the arrival of Nightmare Moon soon after.
In season four, we start off once again with the Summer Sun Celebration. This time, however, Celestia (after being saved along with Luna) announces that this is a special event, as it's the first in a long time where she can cheer the revival of her sister instead of lament her loss. Between the first episode of season one and the first of season four, exactly one year has passed. This is important when it comes to future claims, so keep in mind that everything in seasons one, two, and three have happened within the span of one year.
The seasons in Equestria are handled by the ponies, making each season pass into the next using manual labor, as seen in Winter Wrap Up and Fall Weather Friends. However, both episodes bring about two more important facts: how the seasons are arranged, and how often they occur. I've seen several arguments in various comments saying that the order of the episodes is true to the chronology because ponies simply change the seasons as often as they'd like. This is entirely false.
In Fall Weather Friends, both Pinkie and Spike, when announcing, declare it the "annual Running of the Leaves". Annual, of course, means it happens once a year, making fall only happen once during the single year in which the series takes place. In the same vein, in Winter Wrap Up, the mayor laments that they'll be "late again this year", with organization being what they need to ensure promptness. Again, the phrase assures us they change to and from this season only once per year, making Winter Wrap Up and the only other winter episode, Hearth's Warming Eve, happen within the same winter season. Already, we have early proof that the two episodes break the chronology, but how far does this go?
Before we push farther, it's important to establish a timeline of seasons. As Winter Wrap Up blatantly points out, winter transitions into spring. Fall Weather Friends is a bit more subtle, but because of the lack of leaves on trees during the winter months, we can safely assume that autumn would transition into winter. Already, we have autumn -> winter -> spring. That leaves a single season, summer, to place. Assuming that they only have four seasons (like our world), it either has to come before autumn or after spring, which, as we all know, summer already does both. Thus, the seasons in Equestria match the same seasons here on Earth. The series starts in summer and ends in summer, giving us a very decent timeline to start placing episodes.