How to transition from pure biology PhD to bioinformatics/biostatistics? I've been working as a PhD student in biology and currently in my last year. However I realised just a year into my PhD that I hated doing experiments, I really hated it. I think one of the reasons may be that in the beginning, most of my experiments did not work. Also, I couldn't justify spending years of my life working on a tiny protein that no one cared about, all in the name of basic science. I initially wanted to quit but it was a difficult decision because I am an overseas student and may have had to immediately return home if I did.
During this period, I did find that I enjoyed working on statistics and taught myself to code a bit, mostly at night. I'm currently taking online coding courses and I really enjoy using computational methods within my field especially in statistics. My main question is: is it realistic to switch to a career in bioinformatics and statistics, and how can I go about this immediately after my PhD?
As a PhD student in Math with Computer Science background who is working in Bioinformatics exclusively: it is definitely possible for you! 

In the Computational Biology Institute at The George Washington University where I work as GRA we have PhD students, Post-docs as well as Faculty from different backgrounds (including pure biology) working purely in bioinformatics area.

I do believe you don't have to switch as much, as you think. It only takes a group that is more focused on the "informatics" part more, than on biology. But still caring about Biology to change the main daily routine of yours. And your background will be needed, as bioinformatics can not exists without biological reasoning and biological guidance / insights. 

If I were you, my first step would be to find what is the most "informatics" related problem close to your current research. It might be not your group, who is doing it, but still. Educate yourself a bit about such research by reading related articles. And after that it is as simple as writing a couple of emails to PIs of respective studies and telling them, that you are very interested in collaboration on their topic. This might yield a joint publication or even a post-doc position offer from respective lad.

I wouldn't count much on pure "career" in statistics, as it usually requires more math background. But biologists in bionformatics, who at least knows how to do statistics are very hard to find. So the more you know from the programming / statistics side, the better you are off go to more informatics groups.