Let's take your example if you prayed two raka'as sunnah and two raka'as fard in one go you would have prayed four raka'as which doesn't correspond to neither of both.
Each prayer must be prayed for itself with a own niyyah (intention) and starts with takbeer and ends with tasleem.
The key to prayer is purification; its
beginning is takbir and its end is taslim. (Sunan Abu Dawod and Jam'i At-Tirmidhi)
And for our example we may find in a hadith:
... Then Bilal pronounced the Adhan (i.e. call for the Fajr prayer). The Prophet (ﷺ) then offered two rak`at (Sunna) prayer and went out (to the Mosque) and offered the (compulsory congregational) Fajr prayer. (Sahih al-Bukhari)
which shows the practice as described above.