Not to sound like a condescending confidence poster, but aim upwards. Sure, you feel like you're not the smartest person in the room sometimes, but you can still try to do what you weren't sure you were capable of yesterday. People will automatically assume you're "smart enough" because you're already there. What will impress people is you getting better as they're watching.
Also, everyone loves being treated like they're a full time employee, but remember that you are an intern at the end of the day. This is ultimately supposed to be a learning experience for your benefit. You are being paid for your work, yes. But if a you feel like a company isn't accommodating the fact that you're here to learn and fail at some stuff, that is a valid reason to be concerned and doesn't reflect well on the company.
And finally, having a prestigious company on your resume is pretty great, but it doesn't it doesn't hold a candle to working somewhere where you feel valued and respected. Remember that.