"Four white ponies are running fast."
Translation:Neljä valkoista ponia juoksee nopeasti.
Because when the noun is modified by a numeral, the verb treats it as a singular because it's conceptualised as a single group. If the things being referred to are treated as individuals instead of as a group, for example "all four ponies" as opposed to just "four ponies", then the verb can treat it as a plural.
There are as many plural forms as there are grammatical cases because each grammatical case has a plural form. "Ponit" is the nominative plural form and does not fit because the numeral requires the partitive singular form for the noun that it modifies, which in this situation is "ponia".