This is really interesting.
If you translate "At that time I learned more about Chinese culture." using Google, the result is indeed 那时我对中国文化有了更多的了解 as you said.
However, if you try "At that time I studied more about Chinese culture.", the result will be 那时我更多地学习了中国文化, which is closer to you attempt: 当时, 我学习了更多关于中国的文化, because they both use the word "学习".
I would say, Google translation is very smart to catch this subtle difference between "learn" and "study".
"Study"'s meaning is somehow narrow, so is that of 学习,which is why they are mapped to each other.
"Learn"'s meaning could be wide. Any approaches that provide you with knowledge can be considered as "learning", that means it is too wide to be mapped to 学习, therefore Google uses 了解。
With that said, you attempt doesn't sound 100% normal.
Unless you indeed want to emphasis that you studied in a school or from books, a better translation should be 那时(那段时间), 我了解了更多的中国文化.
(Note I used 那时 instead of 当时,so does Google translation. That is because 当时 refers a very short period of time. Like, 我当时愣住了, meaning I was stunned for only a couple of seconds. But here I guess you want to say you were learning throughout a period of time).