I know sites like fiverr and upworkvexist, but they seem like a race to the bottom where you are competing with people in low income countries that can work for pennies.
What can I do to be able to start picking up freelance contracts within a few years?
The mistake I see many people wanting to freelance/contract do is they think they need to be a generalist in all ways. You do need to be a generalist in terms of technology (at least to some degree) but you need to specialize in a specific area or domain to get started. Your network will grow as a result of you doing good work and being recommended by the small network you develop initially.
If you just go out and say I can do anything for anyone then you are competing with the low cost labor pool overseas. In the end the way you get started is the same way you grow, build a network and utilize it. Posting on forums, or upwork etc is a losing proposition unless you are in a place where you can live on those types of low wages.