European medievalist here! I'm happy to tackle this question with at least a cursory description of the understanding of medical procedure by the Late Middle Ages. I'll be focusing on general medicine at the time period, with a mention of a widely-published surgeon named Guy de Chauliac (~1300-1368).
People who suffered fractures or other serious injuries generally sought the advice of an experienced medical professional in their community. However, the education of that particular individual varied widely depending on the region and the time period. For many, the most qualified source of medical help came from a barber-surgeon. Barber-surgeons were an apprenticed profession. By necessity, most barbers were skilled with razors; their abilities extended to using their tools to handle medical needs. Many were illiterate, but had a practical (hands-on) knowledge of common medicinal practices (blooding, leeching, cupping, pulling teeth, and--if it came to it--cutting flesh). Their profession also extended into the wider scheme of medical or semi-medical services: they joined the ranks of midwives, bonesetters, animal gelders, and other professionals that lacked university training but had an organized hierarchy of experts and apprentices.
In contrast, a university-educated, licensed medical elite existed from the High Middle Ages and onward. At least in Northern Europe, most of these professionals were consultants and researchers; they observed medical procedures and recorded them for academic work, but did not serve patients. Many of these practitioners would have read Aulus Cornelius Celsus' De Medicina, and would have learned about fractures, dislocations, and casts in Book VIII, where he cautions against immediate amputation:
...sed multo melius est ante emplastra experiri, quae calvariae causa conponuntur. Eorumque aliquod oportet ex aceto mollitum per se super fissum fractumve os inponere; deinde super id aliquanto latius quam vulnus est eodem medicamento inlitum linteolum, et praeterea sucidam lanam aceto tinctam; tum vulnus deligare et cotidie resolvere, similiterque curare usque ad diem quintum, a sexto die etiam vapore aquae calidae per spongiam fovere, cetera ex eadem facere.
...it is much better first to try the plasters which are prepared for the cranium. One of these dissolved in vinegar is to be put upon the fissured or fractured bone by itself; next over this, a little overlapping the wound, lint steeped in the same, and over this unscoured wool sprinkled with vinegar; then the wound is bandaged and the dressing changed daily, and so treated up to the fifth day; on the sixth day also the wound is steamed by means of a sponge, then dressed as before.
In this passage, Celsus is referring to cranial injuries, but the advice stood: attempt to heal the fragments before removing them.
Attempts were made to distinguish educated medical professionals from the non-educated: to give an example, the Collège de St. Côme in Paris gave long robes to fully-licensed surgeons, and short robes to those who had yet to pass their examinations. I won't get into the distinction between "surgeons" and "physicians" for the purposes of this answer--but the politics of where "barber-surgeons" and others fit into this scheme get even more complicated.
In terms of procedure, fixing a broken bone was generally the same as it is today: consult with a professional to keep the fracture from shifting or moving, and wait for the body to heal. For leg fractures, the most practical option was usually to rest until it healed.
I'm going to cite a part of de Chauliac's Chirurgia magna (1363), a standard textbook for medical practitioners from the late 14th to even the 18th century. De Chauliac started his medical studies in Toulouse, then moved to University of Montpellier, where he received his magister in medicina (Master's in Medicine) after six years. He later became the private physician for Pope Clement VI in 1343. Clement survived the outbreak of the Black Plague that came through France in 1348, popularizing de Chauliac's abilities. He published his treatise on anatomy, medicine, and surgery five years before his death in 1368.
Most of de Chauliac's treatise covers the same advice as earlier medical advice: keep the fracture still, and let it heal. De Chauliac used previous medical scholarship as a source for his own writing. Consider de Chauliac's writing on broken hands, where he defers to the Arab/Andalusi scholar Abulcasis:
...rectificari docet Abulc. comprimere eminentiam manu supra tabulam...nisi quod apponantur ferula et lino, aut solea & remutetur quarto quoque die. Firmatur secundum iamerium in 12. diebus, & postea ut dictum est, remolliatur.
Albucasis teaches the hand to be compressed on a table...and pressed with a stick [splint] and plaster, strapped and changed every fourth day. Being then strengthened, it consolidates by within 12 days.
De Chauliac's description of splinting and plastering, along with his mention of previous scholars that wrote on it, gives credence to the practice being widespread--even among non-educated medical professionals. With the Chirurgia magna becoming a popular textbook for many literate medical practitioners for centuries, the practice stuck.
TL;DR: hold it still, cover it up, and wait out the broken bone.
Bibliography
On top of /u/samleewrites' extensive answer, I'd like offer a slightly earlier perspective. Bald's Leechbook (named after the individual who commissioned it) is an Old English medical textbook (Leech being OE for doctor) likely produced during the 'Alfredian Renaissance' of the late ninth century. Interestingly, the work contains both a wide array of 'Classical' medicine as well as a vast panoply of contemporary English and British herbalism and what we might term 'folk medicine.' While some of the approaches listed in the Leechbook may draw derision today, it's worth noting that many of them have a genuine scientific basis. The Leechbook acquired a small measure of fame a few years ago when a trial at Leicester University discovered that a treatment it proposed for eye infections was in fact useful for combatting MRSA. Much of the medicine contained therein remains relevant throughout the Medieval period, with similar procedures, treatments and cures appearing, for example, in works like the 14th Century Flemish textbook Yperman’s Cyrurgie.
The phrasing of the Leechbook infers that its reader will both be literate and have previously undergone some form of education in both the preparation of herbs and the making of salves and potions, and at least cursory medical procedures such as making incisions, sewing, setting bones, splinting, bandaging and even excising dead tissue. This does suggest that at least to some extent, there was a profession of trained doctors, although this might well have been simply one role played by a suitable member of a religious community. Within the wider population, there are likely to have been individuals within a community with at least some level of medical knowledge - the "cunning men" of later discourse - a presence further implied by reference in contemporary penitentials to women within a community who carry out abortions for others.
The Leechbook is a great proponent of topical medicine, and contains a great number of salves, many of which can be shown to have had at least some degree of anti-microbial, anti-inflammatory or opioid qualities. For example, in cases of a broken leg, it suggests:
If the shins are broken take bone-wort, grind, pour the white of an egg, mix together for the man with the broken shin.
Bonewort is Bellis perennis, the common daisy, and is known for its astringent properties, having also been used by the Romans.
To actually mend the break, the Leechbook advises:
I.25.3 Wið foredum lime lege þas sealfe on þæt forode lim 7 forlege mid elmrinde do spilc to. eft simle niwa oþþæt gehalod sie gerendra elmrinde 7 awyl swiðe do þonne of þa rinde genim linsæd gegrind briwe wið þam elmes drænce þæt bið god sealf foredum lim.
For a broken limb apply that salve in the broken limb and cover with elm-bark, apply a splint to it, again likewise renew it until it is healed, dispose of the elm- bark and wash thoroughly, then take off the bark, take linseed, grind it to a paste with the elm drink, that is a good salve for a broken limb.
The significance of elm bark is a little unclear. The bark of Ulma genus plants is still used in some cases as a demulcent, although that's largely the American Slippery Elm. The bark of the English Elm could however also have been noted for perceived anti-inflammatory qualities.
So the practice would have been, as today, find the closest thing to medical assistance, clean any wounds, set the bone, apply topical anti-inflammatories and pain killers, splint the wound and then rest.