Welcome to iMore! You probably have your email service set up through IMAP protocol. Basically, what this means is your emails are stored remotely and your phone syncs itself to whatever is on your email server. So if you delete an email through, let's say, a web browser, it will delete from your phone as well. The vast majority of email services use IMAP. Your other option is POP3. Using POP3 to check your mail will download your emails to your iPhone and completely remove them from your email provider's servers. If you opt to use POP3 protocol, which isn't offered by all email services, you generally have to manually set it up through yout phone and there are often extra steps needed to enable POP3 through your provider. I recommend checking with your email service's support for steps to set up POP3. Keep in mind, if you want to ability to check your email from multiple devices POP3 is not the way to go. Since email is being phsyically removed from your email provider's servers, it won't be available anywhere other than the device it was downloaded to.
I hope this makes sense. At the end of the day, most users are better simply not deleting emails they might need at a later time. If you're accidentally trashing email on your iPhone, go to Settings > Mail, Contacts, Calenders, and toggling Ask Before Deleting to ON. This will force you to confirm emails before they're deleted.
Try trashing by swiping left. Maybe it'll work better that way. I get a few that completely delete too. But I like it that way. You can also choose to archive rather than delete. You can change that in Settings.