Originally Posted by MRT2

It can be done, but maybe shouldn't be done, because of the cost of new bars, shifters and brake levers. And, not all mountain or flat bar brake levers and shifters are compatible with road brakes and derailleurs, so you need to know what you are doing.

What I would suggest is you work with someone on fitting your bike. that may involve raising the bars a little, maybe moving the saddle up a bit, or back, and checking the width of your bars. (it is a thing. When I bought my bike, I switched out the narrow drop bars with a slightly wider bar).

Second, you can add a second set of brake levers you can use when riding on the top of your bars.