Can I trademark my logo/company name if I'm only using it in a publicly viewable entertainment website?

Your trademark rights in the "logo/name" of your comic strip will arise all on their own as soon as you start publishing the strip. You don't have to do anything other than consistently label the strip with the logo/name you select. [Before you select one you should have a trademark search performed to ensure that you will not invade anyone else's trademark rights]. Once those rights arise, you can enforce them if necessary (even if the rights are not "registered" with the Trademark Office.

The issue is whether the Trademark Office will register the trademark rights you create in the logo/name. "Comic strips" is a Trademark Office-approved goods description (in International Class 16) -- so that's the description and class that I think applies. The specific issue is whether making a comic strip available for free is a "use in commerce" of the logo/name sufficient to justify registration. I think it is. There's a long rationale for why but that's beyond the scope of this answer. At the end of the day, there's no downside to trying to register the trademark you use to brand your comic strip (other than some money). Good luck.

If you're not using the mark in commerce, which it sounds like you are not doing, then you won't be able to register the mark. You may wish to consider actually using the mark, perhaps by selling prints of your cartoons, in order to establish use in trade.