[1] I run quicker than Sara.

[2] I run more quickly than Sara.

[3] I run faster than Sara.

The usual adverb form has ly suffixation to give "quickly" as in [2], but there is also an adverb "quick", which is formed by conversion from the adjective "quick", and has the inflected forms "quicker" and "quickest". The adverb base form "quick", as in *"I run quick", is unacceptable for most speakers, but the inflected forms are more acceptable than the plain form without ly, which explains why [1] is acceptable for many speakers.

The same applies to [3].

[1] I run quicker than Sara.

[2] I run more quickly than Sara.

[3] I run faster than Sara.

The usual adverb form has ly suffixation to give "quickly" as in [2], but there is also an adverb "quick", which is formed by conversion from the adjective "quick", and has the inflected forms "quicker" and "quickest". The adverb base form "quick", as in *"I run quick", is unacceptable for most speakers, but the inflected forms are more acceptable than the plain form without ly, which explains why [1] is acceptable for many speakers.