何か気になったら気にせず会話中でもどんどん指摘してほしい。って英語でなんて言うの？

*Could you please correct me as soon as I make a mistake.
as soon as means that you want to be corrected when you make the mistake even when you are still talking. You can also ask the teacher to write down in the chat box for you so that you can also practice after the lesson.

If you wish to know immediately every time you make an error of any kind, then you may request that the teacher interrupts you, for example, if you are reading, and then explains what the mistake was.
To pull someone up means to cause someone to stop or pause the activity that they are engaged in:
"The shock of his words pulled her up short."

If you wanted your teacher or tutor to stop you straight away and to correct you when you make a mistake, you could use any of the above mentioned examples. It's good to let them know you would like to be corrected immediately or else most people would wait until you are done, as it can be considered rude to interrupt unless given permission to do so.

When you want to ask your teacher to correct all your mistakes during the lesson even if it means interrupting you; then you may ask in the following ways:
-You are welcome to correct me when I make mistakes in class, even if it means interrupting me.
-Please correct my mistakes during the lesson

"Feel free" is a way of letting someone know that they should not hesitate in their actions, it is a way of inviting your teacher to correct you even if this means interrupting.

All of these sentences are compound sentences that combine the idea of pointing out a mistake and it's OK to interrupt me.

I'd make use of the phrase "right away" and the word "immediately". In common usage, these are the same and mean that somthing should happen very quickly.