Weight vs. Wing length
The approach I'll follow to decide if the Pegasus is possible is to analyze the Weight vs. Wing length. The first problem I have is to find a good enough data source.
In the past (old version of this answer) I had used third party analysis based on Wikipedia data. This time I'll get a better data source and work estimation by myself.
Extracting the data
The data source I found is the information published by the British Trust for Ornithology. In order to extract the data, I've created a custom software. Using that software I got the raw data that I will be working on.
In order to process the data, I used an online spreadsheet.
The extracted data, once curated (removed entries with not enough data) consists of 540 data points corresponding to averages of weight and wing length. This data came from multiple measurements of the individuals per species. I'm disregarding the distribution of those values and only using the average.
Note: if you want to compile the software you need a C# compiler and my libraries available from nuget.
Processing the data
Once curated, I did the following plot of the data:

Plot Weight (?) vs. Wing length (mm)
In the plot we can see that some of the data points lay very close to the horizontal (wing length) axis. The rest of the data points seem to follow growing curves.
Actually the data is wrong, turns out some data is in kg and other in g (my software doesn't read the units) - after fixing the unit we have the following plot:

Plot Weight (g) vs. Wing length (mm)
My next step was to plot the natural logarithm of the values for the same data points. The result is as follows:

Plot Ln(Weight(g)) vs. Ln(Wing length(mm))
In this plot we can see that a linear function seems to be a good enough approximation.
Remember that the wings are at least as big as needed, and not as big as too increase Weight to a point that impedes flight. So, it is expected that for each Weight value, there actually exists a range of plausible wing sizes (the original plot supports that idea too).
Adjust to a linear function

Plot of the first cluster + approximation to a line
In the plot the added line follows the equation:
$$Ln(Weight(g)) = \frac{1640}{1379}Ln(Wing length(m)) - \frac{2102}{2167}$$
Oops... that's the wrong way around - of course, I have weight in the vertical - let me fix it:
$$Ln(Wing length(m)) = \frac{1379}{1640}Ln(Weight(g)) + \frac{7357}{9020}$$
Keep in mind that we are working with the natural logarithms.
Of course, the wings add weight to the animal. So using the weight of the animal without wings is inaccurate. But it should be noted that due to the estimation method, this value is somewhere near the middle of the range of valid wing lengths. Thus, I’ll consider this method to be good enough.
In case of doubt, BTO defines Wing Length as follows:
Wing Length
Maximum flattened chord measured on live birds, this measurement will be greater than that of the natural, resting wing, or of measurements taken from museum skins (see Svensson (1992) for details).
Also, it is a measurement of a single wing; it should not be confused with wingspan.
Horse computing
Subject 1: Thumbelina

Thumbelina compared to a dog.
Thumbelina, the world smallest horse has 26Kg.
So we have:
$$ Weight = 26Kg $$
But we need grams, so:
$$ Weight = 26000g $$
Next we need the natural logarithm in order to feed it into the function we found:
$$ Ln(Weight (g)) = Ln(26000) $$
We compute Ln(26000):
$$ Ln(Weight (g)) = 10.165851... $$
Plug it in the function:
$$Ln(Wing length(mm)) = \frac{1379}{1640}10.165851... + \frac{7357}{9020}$$
Compute that and we have:
$$Ln(Wing length(mm)) = 9.363625... $$
Now we get the wing length:
$$e^{Ln(Wing length(mm))} = e^{9.363625...} $$
$$ => $$
$$Wing length = 11656.574347... mm $$
$$ => $$
$$Wing length = 11.656574... m $$
So, we have Wing length of 11.65 m (38.22 feet).
That's HUGE for poor Thumbelina!
Other subjects
I'll be using data from the paper Estimating a horse’s weight from the Department of Primary Industries of Australia.
Subject 2: Anatomically Correct (Flying) Pony
The large pony has 360kg. Wolfram|Alphing we get: 106.233559... m (348.535 feet)
Ok, that's ridiculous.
Evolution
Before we consider the case of riding the animal, we need to consider that the non-ridden modes of locomotion would have evolve naturally, and then for it to be ridden it would require domestication. In particular breeders would select the stronger animals for reproduction in order to make offspring strong enough for carrying a human.
The Six-limbs problem
As with Anatomically Correct Angels, we run into the problem of six-limbed mammals. There are no examples in nature, and there reason to believe that the DNA structure doesn't support it※ - of course, a different structure could also evolve.
※: Researchers have created mutated fruit flies that have legs where they should have antennas. This was done by swapping portions of DNA, this suggests that there is a limited number of locations where you can insert limbs in the DNA.
It is also worthy to look at the lore:
Bellerophon on Pegasus spears the Chimera, on an Attic red-figure epinetron, 425–420 BC
The depiction above gives some insight on the six-limb problem as we can imagine the wings as ramifications of the frontal legs.
I also want to note on the wing aspect ratio, that different wing aspect ratios are characteristic of different flight styles:

Examples of Wing aspect ratio
If Pegasus is used for battle, it probably has high agility as to avoid attacks and approach the target from different angles. So, a wing with similar proportion to the crow depicted above example is expected.
Conclusion
Giving wings to horses result in exaggerated size
There is no way around it, if want something that resembles a Pegasus, you need to evolve the horse to have less mass. Similarly to the Anatomically Correct Angel, the Anatomically Correct Pegasus needs to have thinner bones...
The Anatomically Correct Pegasus has two solutions:
It can fly but you can't ride it because the bones are too weak to support a human.
You can ride it, but it can't fly, the wings are only an ornamental vestige kept by the breeders.