No You are mistaken, Qt apps. are not big at all.

For example how much space uses windows notepad.exe? Application is like 64KB but with all dll's it's over 10MB.
Only difference is that all notepad dll's come with standard Windows installation. So one can say blame M$ for not including qt dll's in windows .
Simple remedy to this problem is copying libs into %windir%/system|system32, or better adding into PATH variable Qt lib location, so You don't need to copy dll's to apps location.
But seriously it's normal and You just see that because those dll's are not present by default in Windows.

Qt by default is build as shared because they are licensed as LGPL. With static You go with commercial or GPL license and share your src.
Shared is better because program (ironically in Your post context) is smaller . Qt is not present in Windows but is by default present in KDE Linux base environment (KDE is build using Qt). So If you want to ports Win app. to KDE/Linux You will need all Windows dll's and situation will be in reverse (and is worse then that, because probably all Win dll are proprietary, so porting is not so easy - that's why we use Qt to port apps. easily across various OS).

Also statically linking won't solve size problem (at least not without changing default Qt build specs)

To shrink app You could compress it (upx).