All web addresses begin with either HTTP or HTTPS. Because HTTPS is the more secure option, you may have links to a site that switches to it from HTTP. That's a good thing for keeping your audience safe — but it also may affect your short links.
In order for your links to point to the correct version of the destination URL, the owner of that site has to implement a rule redirecting all traffic to use HTTPS. The good news is that this redirect is very common.
For example, let's say you have a short link mykicks.co/my-cool-site that points to the destination url http://mycoolsite.com.
If the owners of mycoolsite.com switch to using HTTPS, your link needs to redirect to https://mycoolsite.com. This only happens if the owners of mycoolsite.com have implemented the HTTP-to-HTTPS redirect rule.
A website's owners are the only ones who can control whether its URLs use HTTP or HTTPS. This isn't something that can be set by you or by Bitly.