Dude...I would honestly be kind of off put by the girl if she did this. Its literally been four dates. Hell...it sounds like you haven't even had sex yet. She has no stake on your dating life, who else you're seeing, what you're doing etc. She's completely in the wrong here. If you want to continue seeing her, maybe being directly confrontational won't help. Because that would probably just spark an argument.
But had you not yet lied, the solution would have been pretty simple. You should have told her that yes, you are on other dating apps. That doesn't mean you're not serious about her and that you dont' like her. Rather, it just means that you like to take things slow, and get to know someone before you commit to dating her exclusively. Maybe don't phrase it exactly like that. But basically, you want to reassure her that you're into her and that she's the one you're looking at rn etc. But that she also needs to respect the fact that this relationship is still young, and that you still have your autonomy.
Regardless of how you phrase it, she may not be ok with this. Some girls have this unreasonable expectation that you'll date them exclusively, regardless of weather or not its an official relationship. In which case, she might break it off. And that's fine. But more likely than not, if you stayed calm and in control, but also warm while you explain this to her, she would have realized that you're right.
BUT...
Now that you've lied to her and tried to hide it, you look very weak. I don't really know how you can effectively recover now. Because you've officially given into her frame. By lieing to her, than getting caught, you've communicated to her that "woops, you're right. I shouldn't have been using dating apps. But I did anyways". Which means, that it's a lot harder for you to tell her that its unreasonable for her to expect you to be off them after four dates. And you were also dishonest with her, which is never attractive.
Having said that, here's what I would do:
Next time you see her, if this comes up again, say "look, I really like you. And, to be honest, the reason I lied to you is because I was afraid I would lose you. That's why I told you I'm not on other dating apps. But here's the honest truth...I don't really intend to meet anyone else off those apps for the near future. BUT, I like to take things slow in a relationship. We've only been on four dates, and so I assumed we have no commitment to each other yet. Do you see it otherwise? She'll think about it and say "I guess not" or something along those lines (and if she says she DOES see it differently, then you need to talk about that and figure out exactly how she sees it and where your relationship is at). So then you'll continue on..."yeah, that's what I was thinking too. So that's why I'm still on those dating apps. When the time comes, I promise I'll delete them. But I just need to know that this is what we want first. Why did you think I shouldn't be on a dating app?".
^
This reassures her that you're very into her, and that she's not just another girl in your rotation of hoes. It also shows that you empathize with her and care for her perspective (all of this fixes the attainability problem caused by being on online dating apps in the first place) But it's also firm. It lets her know that you still have boundaries and that she needs to respect your freedom (helps fix value problem caused by your weak frame. Still won't be enough though). Finally, that last part where you ask her "why did you think I shouldn't be on a dating app?" turns it around on her. It forces her to defend her position, making her look like she's the one in the wrong (which she honestly is). Also, when you force her to think about it and give a rational explanation for her position, she'll probably realize that she has no legit reasons other than her emotions for why you shouldn't be on dating apps (given that she's a reasonable person. It's also possible that she'll realize she's wrong and lash out at you in frustration, in which case I would reconsider dating her altogether). Cause I guarantee you that pretty much any answer she gives you will be stupid. Then all you have to do is point that out to her (don't tell her that what she's saying is stupid! Point out what the flaws in her reasons are). She will acknowledge that you're right, and you can move on.
Keep in mind that while you say all this, you must keep a warm vibe towards her the entire time. She needs to understand that you're not attacking her. Rather, you're simply trying to express how you feel about the situation, and that you're trying to understand her perspective. Otherwise, she will either go into either auto-reject, or she will start a verbal argument, or both.
Given the bad frame you've already set, there's a good chance even this won't work. But, something along those lines is probably your best shot.
The lesson here is...don't do this again. If you think you're perfectly allowed to be doing something, don't apologize for or hide it just because you think it will upset her.
