I can assure you that your turn based 2D games can use the advantages of C while not being able to take full advantage of some performance characteristics of C++ an C#, yet not needing them for most 2D games. Once you get large libraries of code and many art assets in a couple years, then performance might get critical - or maybe not.
A small correction is needed here as you seem to be assuming that C++ and C# have a better performance than C which is not the case at all (especially in the case of C#). The reason the majority of the industry shifted from C to C++ was do to it being easier to maintain larger codebases with multiple people working on them, and requiring less lines of code... so it was to save time and energy and had nothing at all to do with performance. As the C language is basically an "easier to read" version of assembly you aren't going to get any faster.
After a year or two, if you then decide to start making 3D games or your 2D games have been filled in order to need the extra performance, then consider C++ or C#. (Who knows: Maybe Java or C will be in the league for the highest performaning games again in a few years.)