There is only one way to toughen up: Practice. Be it interviews or sports. Be it negotiations or cooking. Practice and practice perfectly.
1. Take mock interviews from experienced engineers
Go one topic at a time e.g. Recursion or Trees or DP or Multithreading etc. Before you walk into the interview, practice well. Do at least 20 problems on that topic, if not more.
2. Do practice interviews at companies you don't want to work for
Don't feel an imposter in doing this. Realize that companies also go thru several candidates before they pick one. They are normal people like you, they have a learning curve to fill the position, and so do you. Finding the right company, at the right stage of your life, is not a trivial thing to do; it's a project and needs to be managed like that.
Going thru various interviews hardens your skills of negotiation, your understanding of business models, and more importantly, gives you clarity about what your strengths and limitations are.
3. Practice the right mindset
Never go to interviews with a loaded agony of getting a job. Go simply with the intent of solving interesting problems with interesting people ("to add life to your character"). Nobody explains this better than Bryan Cranston (possibly one of the most successful TV actors of recent times): https://www.youtube.com/watch?v=v1WiCGq-PcY
I make these observations based on my experience running a successful bootcamp for interview prep (http://interviewkickstart.com) and also being an early engineer and a hiring manager (Director of engineering) at Box prior to that.
[Edit: specifically mentioned Bryan Cranston by name, based on the comment below]