What versions of the rsconnect and curl packages do you have? Current versions are 1.3.4 for rsconnect and 6.2.0 for curl. Some time back the names for curl options changed, so if you have a mismatch between rsconnect (newer) and curl (older) that might be the source of the error.
I'm not sure about the relationship between RCurl and curl, but I do not have RCurl installed (I do have curl) and rsconnect seems to work fine on my system.
I noticed that your locales were set to Turkish. If you don't want to start RStudio that way, you could check whether changing one or more locale settings to "C" (or English) in your R session fix the issue. See "help locales" for the details.
Frankly, I'm surprised that language settings would cause that one option (only) not to be recognized. It might deserve a bug report.