How can I convince my supervisor that I have enough data to support my findings?
I don't think that anybody can possibly provide an answer without having access to many more details. The precise field matters, how well established the prevailing theories in your field are, how the quality of your data compares to previous work, how the rigor of your arguments compares to other work, etc.
Generally, making these sorts of judgements is what your advisor is for, because he/she has many years of experience in the field.
Now it can happen that your interests and those of your advisor are not perfectly aligned. Your advisor may be too conservative, because he/she does not want to make a mistake. You may be overly aggressive, because you want to get your thesis done.
The question is how to proceed. You could ask your advisor to let you finish a draft, and show it to a third person, an expert in the field (possibly a member of your thesis committee) with no direct interest in the outcome, and who you can trust to keep a confidence.