Which is the most normal translation:
Я хочу осушествлять или реализовать мои мечты
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Which is the most normal translation:
Я хочу осушествлять или реализовать мои мечты
Я хочу осуществить свои мечты. Don't forget about the proper verb aspects and all.
Я хочу реализовать свои мечты. This one is fine.
As for which would be most "normal"... They're both absolutely fine. No drama here. The first one might be a little bit more "Russian" :) Therefore you'll hear it more I guess.