As it has been pointed out by other answers to this question and this related question, there are ample examples of historical left-wing conspiracies. Why this has changed recently to being conservative-dominated I would posit can be attributed to two factors: A steady decline in conservative trust of institutions, especially major media outlets and the changing demographics of the Republican party
Distrust:
Republicans have markedly low trust in academics and government workers, and both sides distrust elected officials and business leaders. This comprises essentially all institutions outside the state security apparatus and religion. The lowest trust is in journalists, who disseminate the information generated by the institutions that are already distrusted. If you don't trust any of the people saying that Democrats are not stealing and drinking childrens blood, the belief will continue.
While Republicans do have media sources in Fox news, and to a lesser extent opinion radio and Newsmax/OANN, Republican trust of media has been lowering for decades, with a steep drop at the start of Trump's term
Fox news is presumably not considered "mass media" by respondents despite being the most viewed news source in the US, unless a significant amount of Republicans watch but do not trust it.
Changing Demographics
It was the case 30 years ago that the Republican party had higher educational attainment than the Democratic party. This has seen drastic change - Trump's strongest demographic was white men without a college degree. While educational attainment is not necessarily synonymous with intelligence, most conspiracy theories do not take a large amount of intelligence to dispel. Clearly this has been lacking.
Conclusion
Republican distrust of all institutions outside of religious leadership and police/military (and implicitly Fox news) combined with low educational attainment sets the stage for conspiracy theories to flourish, and social media provides platforms where such theories can spread instantaneously and be reinforced by peers.