Hello Hsottil,
Thank you for using Apple Support Communities.
I see that when you enter a new contact on your iPhone/iPad the change isn't reflected on outlook.com or gmail.com. And the same thing will happen if you edit a contact. I understand the importance of being able to add and edit your contacts so they will be updated.
First, please make sure that both of these accounts are set up on your iPhone by following this path: Settings > Mail, Contacts, Calendars, Accounts. If they are already set up, select each account, and make sure that Contacts is enabled (green).
If they still aren't syncing, please give your iPhone a fresh restart, and then test again. The following article will help:
Restart your iPhone, iPad, or iPod touch
Cheers.