"I go out to the hospital every week."
It is not normal Japanese to "dekakeru" to the hospital. Dekakeru is usually used for fun activities. Normally this sentence would be expressed using the regular word for to go, "iku", not dekakeru. One might say one "dekakeru" from the hospital, because leaving it, you are actually going out of it, but one would not normally go out to a hospital. Dekakeru would be going out with friends to have some fun
Interesting nobody called bullsh#t on his comment, you can literally google it and find natives using it like this, lol.
病院に出かける and 病院に行く are two different things. The former implies you are going out of your house to do something there, the latter just describes the trajectory of "going", you could go to the hospital from the park for example, but you go out to the hospital from your home, probably to get checked or to visit someone.
Perhaps no one has called ❤❤❤❤❤❤❤❤ on this and it has 27 upvotes over the past year because the native speakers who peruse this forum agree that my comment is correct. That is not to say I am an authority--not a native speaker, and I have only lived in the Kansai area for 6 years, so other areas may use it differently, but my wife is a native speaker and I checked with her about this before posting. That is not to say you can't find something that disagrees with me on the internet. You can find anything on the internet.
In English if you say you are going somewhere, that means something different usually than if you say you are "going out." I tell someone I am "going out" tonight, they think I am going dancing, or to a movie, or to a restaurant. I think dekakeru has that same connotation in Japanese. Feel free to disagree, but I thought it useful to mention to other duolingo readers that if my wife was going to the hospital, say because she is sick, or visiting a friend, or maybe going to work there, she would in all these cases not use "dekakeru", but "iku". She would use dekakeru if she were going to meet some friends and go shopping or head to a yakitoriya or kissaten. Native Japanese speakers, please feel free to correct me if I'm wrong.
that's fine dude, next time just state your opinion, I know English is hard but you could at least start with "My wife thinks..." or "I think is better to use...". "It's not normal to X" is an absolute, not an opinion, and one would think that someone that knows Japanese would know better about being humble.
I do respectfully disagree with you, because while you are in part right about how 出かける is often used, that doesn't have anything to do with the word itself, it is just more common to say you are going out to do recreational stuff. When you have to go to the hospital, the phrasing just doesn't go well with it because it would imply that it is a decision on your part, while in reality, most of the time "going to the hospital" is a need.
I would personally use 出かける for a case where I have to make an appointment for my friend (as a favor) or when going out to visit a friend in the hospital, in that case, is more like an errand than a need and it would make sense. If I wanted to say I go to the hospital every week to get myself checked I would just say 週に一度通院しています。