oh-my-zsh does not start on mac

I had a similar issue but I resolved it by executing the command chsh -s /bin/zsh in the terminal,then navigated to

Terminal -> Preferences -> General window

and changed the Shells open with: option to Default login shell while the command (complete path) set to /bin/zsh.This should solve your problem.

Also, I tried to change my default shell to zsh: [~]$ chsh -s /bin/zsh

try

chsh -s $zsh

that worked fine for me

You have to change the default shell to zsh.
To do so use the following command.

chsh -s /bin/zsh
restart your terminal

I took it from here.
These were the steps I followed when I installed zsh.
I hope it will help you as well.

Not only in System Pref>User and Group>...>User profile's shell settings, but also in Mac's terminal>Preference>ChosenProfile>Shell section, I had to change /bin/bash to /bin/zsh. after that, it works! Thank you, @RemyJ!