Only PNG supports transparency, is that true?

• png-32 supports different levels of transparency. Each pixel can have an opacity between 0 and 255, with 0 as completely transparent.

• png-24 supports setting one color as fully transparent. Everything else will be opaque.

• gif uses a color palette. You can specify that one color in the palette is fully transparent.

• png-8 also uses a color palette but each color in the palette has its own opacity value, so it supports varying levels of transparency too.

• jpg does not support transparency.

JPEG XR (not the same as JPEG 2000) also supports transparency.
However it is not a useful format for the web as only Internet Explorer and Edge support it.

WebP is another emerging image format with support for transparency.
However, Internet Explorer lacks support for it.

JPEG 2000 supports full transparency. See JPEG 2000 Side channel spatial information.