"You are welcome, goodbye!"
szívesen is literally something like "from the heart".
It is not a literal translation of "welcome" -- but Hungarians use szívesen in a context (such as replying to thanks) where some English speakers might use "you're welcome" or "don't mention it" or "it's a pleasure". But it's not a literal translation for any of those.
Treat it as a phrase of its own, equivalent to "you're welcome" or whatever you say, not as an adjective that means "welcome".