Ortiz was a 235 hitter and on his way out of baseball when Boston picked him up. Good choice by them, but question marks abound concerning the marked improvement.
Fun cherry-picking of a stat. He batted .272 his last year in MIN. Overall, even with the .234 the year before that, he was a .277 career hitter when DFAed by MIN. And Hrs went from 10 to 18 to 20 those last 3 years.
Can't speak to whether he took anything or not in that timeframe, as there's nothing to say he did or didn't. Be shocked if he was completely clean, I guess, but who was? Never failed a drug test, but did show up on the Mitchell Report. (Tested for more than just steroids, by the way, and didn't say who was on there for which substances).
But since you pointed out how crappy Ortiz was before Boston, it'll please you to know that after 2003 is really what they're calling the Testing Era. So pretty much all the stats you want to point to as why he'd be worthy are in an era where Ortiz was tested for roids. If his big numbers were before the failed test, you'd have something, but all the good stuff came later.
Not to say people still aren't doing something, as some people always stay ahead of the system, cheat in the off-season, whatever, but Ortiz hasn't failed a test during his decade-long run of dominance...
Always a few players that take longer to figure it all out and burst onto the scene a few years later on, after being tossed around a bit. Jose Bautista has played for 5 teams, and was middling and almost done when suddenly became a force in Toronto. Chris Davis was doing nothing and almost done when suddenly in 2012 got it going, and blew up big in 2013. Texas gave Adrian Gonzalez to the Padres in a trade because he looked like he wasn't going anywhere. Turned out ok.
Just saying, it happens. Won't speculate whether all of that list (or any) was all-natural, but plenty of examples, and that was without even digging any, just first couple that popped up in my head.