Complex sentence: Despiau began exhibiting at the Salon des Artistes Français, from 1898 to 1900--then moved to the Société Nationale des Beaux Arts, where he showed from 1901 to 1921, and finally to the Salon des Tuileries, where he exhibited from 1923 to 1944.
Split into simpler sentences: Despiau began exhibiting at the Salon des Artistes Français, from 1898 to 1900. He then moved to the Société Nationale des Beaux Arts, where he showed from 1901 to 1921, and finally to the Salon des Tuileries, where he exhibited from 1923 to 1944.