Where did prejudice against left-handed people originate from?

Although this is an enormously broad question in that we are talking about thousands of cultures across thousands of years, the prejudice against left-handers has been fairly widespread. If the answer had to be boiled down to one or two words, those words - for a variety of reasons ranging from necessity to social pressure to belief systems to practicality - would most likely be 'conformity' and 'normality'.

Nonetheless, there are important qualifiers and exceptions which cannot be ignored. Discrimination against minorities probably dates back to the beginning of time but minorities have also been subjected to varying degrees of discrimination at different periods of time and in different areas of life, and thus it is with left-handers. This bias, as noted by the OP, is evident in languages:

The words left and left hand in almost all the world’s languages are synonyms for “defective” or “sinister.”¹⁹ The word for left in Latin and Italian is sinistra, in French it is gauche, and in English left comes from the word lyft for broken, while in German, linkisch is associated with awkwardness.²⁰ Labeling someone as lefthanded (levja) in Russian is a metaphor for deceptive or untrustworthy.²¹ In Mandarin (Chinese) the character for left, zoû, is variously translated as weird, unorthodox, wrong, incorrect, different, contrary, or opposite.

Source: Howard I. Kushner, 'Other Hand: Left Hand, Right Brain, Mental Disorder, and History' (John Hopkins University Press, 2017)

Note also that the percentage of left-handers in different time periods and cultures is often hard to determine. This is not only because we lack data before the modern era (note, though, this research on cave painting) but also because the data we do have is muddied by a reluctance to admit to left-handedness in some cultures and, until fairly recently, the forced use of the right hand. Further, people are left- or right-handed to different degrees.

Professor of Science & Society Howard I. Kushner, in Other Hand: Left Hand, Right Brain, Mental Disorder, and History (2017), while noting that there are societies which are exceptions, summarises that

...the history of left-handedness parallels that of disabilities. Thus, left-handedness was understood as a sign of abnormality and of the profane, which was opposed to the normal or the sacred.

...toleration of left-handedness serves as a barometer of wider cultural toleration and permissiveness. Societies and cultures that discriminate against left-handers are also less tolerant of other forms of diversity.

Thus, there have been variations in the percentage of reported left-handers in cultures: the more conformist the culture, the lower the percentage of reported left-handers.

Practices resulting in the reduction of left-handers in a population have been referred to as “the cultural pressure hypothesis.” Psychologist John L. Dawson found support for the hypothesis in his comparison of the Temne of Sierra Leone with Arunta aboriginals of Central Australia. The Temne culture, being more conformist, reported a 3.4 percent left-handed rate, compared with the 10.5 percent reported among the more permissive Arunta.

Source: Kushner

Dawson's hypothesis, where obedience and conformity are highly valued, is not the only explanation which has been forward for the regional differences, but it perhaps has more supporting evidence than the genetic one which is complex and hard to prove. The genetic theory is partly undermined by the notably higher incidence of left-handed writers among Americans under the age of 30 (15%) compared to those over 65 (6%); when the latter were children, it was still not uncommon for left-handed children to be forced to use their right hand. On the other hand, left-handed parents are more likely to have left-handed children.

Conformity, of course, has practical advantages for a society and was essential at times. For example, in the Greek phalanx, the shield had to be held on the left arm to protect the man to one's left. A left-hander had to adapt to the majority or the phalanx would fall apart in combat. Thus, conformity was a necessity for survival. On the other hand, left-handed gladiators held an advantage over right-handed opponents, but they invariably fought as individuals. Conformity, if anything, worked against these individuals when in the arena.

Religion has also played a large part in the bias against left-handed people:

Judeo-Christian religions attributed the left with femininity and inferiority, as Eve appeared on and developed from Adam’s left side.

Evidence of bias against left-handers in the Old Testament can be interpreted in different ways, but it's probably fair to say that it mostly (and simply) reflects the prevalence of right-handers in society - so biased, yes, but not to the extent that became evident later. The more stridently biased views against left-handers in Europe, the Middle East and north Africa seem to stem more from the middle ages in all three of the main religions. Among Catholics,

...left-handedness was vigorously oppressed in medieval Europe, albeit not in any systematic way. Left-handers were routinely accused of consorting with the devil...

while in Islam,

Muslims must handle food and write with the right hand. Using the left hand in public and greeting another person with the left hand is a sign of disrespect.

Rather than lessening with time, the prejudice against left-handers became very pronounced in 18th and 19th century schooling with children forced to write with their right hand. As TheHonRose points out in a comment, this persisted well into the 20th century.

Despite the limited reforms of the Age of Reason and the Age of Enlightenment, the 18th and 19th Century were particularly hard on left-handers, and discrimination against them became engrained and institutionalized. Even in the relatively free societies of North America and Western Europe, deliberate and sometimes brutal attempts to suppress left-handedness and impose conformity in the education system were endemic during this time, including such practices as tying a child’s left hand behind his chair or corporal punishment for anyone caught writing with the left hand.

Religion had much to do with this, but it was not the only factor. Left-handers need to be taught writing differently, something perhaps not easily grasped by a right-handed teacher.

Languages that are written left-to-right, like English, are more difficult to write with the left hand -- a right-hander writes away from his body and pulls the pencil, while a left-hander must write toward his body and push the pencil.

Anti-leftism was further reinforced by the views of the Italian criminologist Cesare Lombroso (1835-1909) who

identified left-handedness as a mark of pathological behaviour, savagery and criminality.

However, the anti-left bias has not been evident throughout all cultures, and some cultures recognized both positives and negatives to left-handedness. For example,

The ancient Celts, for instance, associated the left with femininity — the source of all life — and thus worshipped the left side, treating it as sacred.

Also, in China, Kushner and others have noted ambiguities concerning the superiority of the right hand. For example,

In ancient China good fortune was attributed to the left hand. “On occasions of festivity to be on the left hand is the prized position; on occasions of mourning the right hand,” wrote Lao Tse, in the most widely read Chinese book in the world, the Tao te Ching (ca. 6th century BCE).

Source: Kushner

Plato argued strongly against the negative attitude towards the left hand:

The negative attitudes toward left-handedness, while persistent, were not universal. In The Republic (380 BCE) Plato rejects the popular negative characterization of the inferiority of the left hand and condemns the practice of restraining its use. Conceding that the right and left hands have different tasks, Plato points out that, like the two feet, the two hands are complementary: “In natural ability, the two limbs are almost equally balanced; but we ourselves by habitually using them in a wrong way have made them different.”

Source: Kushner

Other source:

Daniel M. Abrams and Mark J. Panaggio, 'A model balancing cooperation and competition can explain our right-handed world and the dominance of left-handed athletes' (Journal of the Royal Society of Interface, 2012)