Activating more type accents on macOS
The easiest way to type Turkish is to use one of the Turkish input sources supplied by Apple.  Make sure you have the Turkish flag showing in the top right corner of the screen (just adding it to Input Sources preferences is not enough).  To see which key does what, use the Keyboard Viewer.

Another way is to add the input source called ABC Extended to the list of Input Sources and then select that in the "flag" menu at the top right of the screen.  You can find the codes for all the letters you need here.

You can pretty easily add your extra characters to unused keys on the US layout (or add brackets, etc to the Turkish layout) with Ukelele.

It may be possible to add characters to the Press/Hold popup menu of the US keyboard.  For info on that see this earlier question.