If you want to average 20+ mph over a triathlon on a road bike without clip-on aerobars, you're looking at 200W of output over flat-ish terrain. This is going to take training, but should be doable for most people with training on a reasonable road bike with reasonable bike clothing, at least on shorter distance triathlons. Doing that over a half-IM or full IM distance is more demanding, and a TT / Tri bike will pay dividends there; the two IMs I did on a normal road bike without clip-ons I averaged 33,7 km/hr and 33 km/hr on 200W and 199W average, respectively; the guys doing the same power on TT bikes who are of my approximate size were about 2 km/hr faster at the same races. If you find you really enjoy triathlon and want to do a lot of it, it's worth considering - there's no prize for using inferior gear and they can be found on the used market in good condition - if you just do the occasional triathlon for fun amongst other things, though, a road bike is entirely fine.

A set of clip on aerobars are a cheap way to split the difference between a road bike and a TT bike (which are, however, only allowed in non-draft legal triathlon, and where I am all triathlons shorter than half and full IM are draft legal), but you might have to adjust the fit to the point it becomes uncomfortable to use without them. I personally find a TT bike to be much more comfortable than a road bike with clip-on aerobars, which is why I don't put them on my road bike (which I also ride granfondos and such with, where you can't have clip-ons obviously, and I don't want to mess around with fit when I'm changing configurations). I find I need to have my handlebars higher to comfortably use clip-on aerobars.

Staying within the realm of "normal road bike", some things do help get more speed out of the same power; position adjustments are a good one, although they aren't entirely "free", because you're constrained by comfort - inch the position ever so lower until it becomes uncomfortable on a long ride (100km+) then inch it up a bit from there. Revisit it in time as you get more used to the position, but you'll find everyone has a limit to the position where they can perform in. Extreme positions are useful for descending, but being bent over silly is going to do you no good if you can't produce power in the position. Narrower handlebars also slightly help but can cost a bit and it's a bit of a faff to change them. Top shelf tires also help and don't cost very much over cheaper versions - don't grab winter tires or a pair of gatorskins or something. If you want a low cost (relatively) improvement, buy a good and properly sized trisuit; properly sized means you can just about zip it up - this avoids any loose fabric, wrinkles, and so on. I'm a big fan of Castelli's trisuits; at 179-180cm and 73kg I wear a size S. It stretches a bit when I put it on, but is supremely comfortable, wicks well, the pocket arrangement is great. The little things do add up to measurable differences, and don't cost very much.

Oh, and do buy proper cycling shoes and pedals and learn to ride with them; they won't improve steady state power, but they'll improve your sprint and sub 1 minute efforts, they'll improve your comfort when you find a good pair of shoes, the control is better, they're just worth it. They last an awfully long time; the cleats not so much, but if you avoid walking much on them or dragging them on the floor, they last fairly long, too. I ride my gravel / commute bike with flats because it's more convenient around town, but if I'm going to ride 100K+ I'm putting on my cycling shoes.