So afaik canon doesn’t explicitly say that Prowl ran the New Institute but…
Prowl provided Skids and Getaway with New Institute technology:
The shady stuff Prowl got up to with Mesothulas seems pretty similar to the shady stuff we know the New Institute got up to (not to say Mesothulas was officially New Institute, but he could’ve been the precursor):
Prowl knows what Chromedome did in his “past life,” which if it’s referring to the New Institute, happened after they had their falling out (this could be about something that happened before that, but we know Chromedome was up to shady shit at the New Institute, and we have no reason to believe he was doing anything shady before that, so I’m comfortable with this assumption):
And the thing I find most convincing but also definitely missed (thanks to @gaymilesedgeworth for pointing it out): Cereberos, who worked at the New Institute…
…called Prowl his “ex-employer”
So in short, Prowl definitely ran the New Institute.