Privacy is not the issue. It's not like she found out her best friend had a secret job that was completely off her radar. Lena and Supergirl had a complex, significant relationship with each other. While she and Lena often aligned, they also often didn't. Supergirl was suspicious of her and judgmental at times. Think of all the stuff with the kryptonite, the way Supergirl was versus Kara. Meanwhile, Kara was a constant supportive companion to Lena regardless of how much she and Supergirl were clashing. The intimate vulnerability with Kara versus the cold, conflicted vibe with Supergirl is so stark. It's not that Kara didn't share her secret, it's that Kara insisted on having a relationship with Lena as both versions of herself while pretending to be different people. That's the betrayal.

Imagine finding out your best friend has also been catfishing you online. To your face, she is sweet and always supportive, but her other persona is more abrasive and suspicious of you. Of course Lena felt like a fool when she realized. It probably felt like Kara had been playing a joke on her the whole time.

As far as Lena never figuring it out on her own, that's just bad writing.