How do I stop notifications at the top of the screen while I am using my phone

When I open my phone and start navigating around the phone gives me previews of all of the stuff I missed like texts, emails and more. If I swipe up on an email alert/notification it goes away only to be replaced by a preview notification of a text I received and so on and so forth. This goes on until I am up to date on all of the stuff I missed. The problem is that if I want to say, search for an email in my gmail, I can't access the search bar because it is covered up by the preview and when I swipe up I have to tap the search bar before the next preview/notification pops up or I will accidentally tap the preview and get sent to another app on the phone. I'm sure this is something simple to stop but I can't figure out where to go to stop these. I still want to see notifications and previews on my lock screen. Thanks for any help!