On a slightly less dramatic note, you have not, as far as the company you are contracted to is concerned, proved yourself.
Particularly as you are normally on a relatively short fixed term contract, their view is usually "why take a chance on an unknown quantity when they can already have someone with the 4 stripes walk in the door without the risk"
One could argue that after some time there doing a good job in the RHS you are now a known quantity ,"better the devil you know etc", however few companies seem to see it this way,rightly or wrongly.
Of course there may be some "dodgy" countries/companies that will bite your hand off with your total experience, & previous turboprop command,and whip you into the LHS overnight. If you choose to go that route take care, but it may indeed get you the jet command experience you seek ,if you are happy to take the risk.
More reputable companies, I don't see it happening frankly.