This has been a myth that proved extremely hard to stamp out. We finally know testosterone does not fuel or increase your risk of prostate cancer. In 1941 a man named Charles Huggins conducted this study basically "proving" that testosterone does in fact fuel or cause prostate cancer. So for 70 years men with prostate cancer received chemical castration treatment for prostate cancer.
Fast forward to 1996 and Dr Morgantaler from the University of Harvard produced a study of his own that totally contradicted what everyone had originally thought for 70 years:
Shifting the Paradigm of Testosterone and Prostate Cancer: The Saturation Model and the Limits of Androgen-Dependent Growth
He basically took 4 men with stage 2 and 3 prostate cancer and injected them with testosterone. Turns out all of the men responded positively to the treatment. The medical community was in shock and many rejected the new claims citing that it was not a large scale study.
Shortly after in 2008 the Journal of the National Cancer institute pooled data from all over the world including 18 different studies and over 6,000 men. The study examined the likelihood of developing prostate cancer in relation to the concentration of hormones in their bloodstream. No correlation of prostate cancer was found whatsoever with any of the hormones studied. Including: TT, free roaming testosterone and other androgens.
Conclusion: In this collaborative analysis of the worldwide data on endogenous hormones and prostate cancer risk, serum concentrations of sex hormones were not associated with the risk of prostate cancer
Endogenous sex hormones and prostate cancer: a collaborative analysis of 18 prospective studies
Here is another more recent piece of literature from 2014:
The long-held belief that PCa risk is related to high serum androgen concentrations can no longer be supported. Current evidence indicates that maximal androgen-stimulated PCa growth is achieved at relatively low serum testosterone concentrations. It may therefore be reasonable to consider testosterone therapy in selected men with PCa and symptomatic hypogonadism.
A New Era of Testosterone and Prostate Cancer: From Physiology to Clinical Implications
So the answer is no testosterone does not fuel prostate cancer. This is just the tip of the iceberg too, there is a plethora of evidence showing that not only does TRT not fuel prostate cancer but it may actually help in early detection. If high testosterone levels contributed to prostate cancer there would be many 21 year old men that had it.