begin -> began

No accordance of time, main clause is in simple past tense, subordinate clause is in simple present tense. So subordinate one must be in simple past tense.

İf you know English grammar,you will see that there is a noun clause there. İf you ask a question with "what" you will see the answer: What did George feel? He felt (that)the train began to move. First the movement started then he felt it.
On the other hand, I know "begin" is present form/bare infinitive.

You don't need a native speaker to confirm what you have been saying. I agree with you. But there is a slight difference between the patterns with -ing form and bara infinitive form;
For example: I saw the woman cross the street.(The whole action finished)
I saw the woman crossing the street.(The action did not finish)
Let's come to the point;
George felt the train begin to move or George felt the train beginning to move. Which one is correct?
Did George feel The train's movement after beginning or while beginning. In my opinion,while beginning. So we had better say; George felt the train beginning to move.
I took "feel" as "understand" in the sentence that I corrected. Thanks for your explanation. Take care of yourself.