First, don’t resort to the tire irons to get your clincher tire seated in the wheel. The chance that you’ll maim the inner tube is too great.
I know how hard it is to refrain. I do, because guess who has two thumbs and has a tough time not reaching for them when the going gets tough (and did maim the tube)? This guy (points both thumbs back at me).
First things first, if you’ve got room (2014 or newer road bike, you should), use 25 or 26mm tires. They’ll give you a little more play, they’re not noticeably slower and they vastly improve the ride of your bike.
Second, get the bead to the center of the rim by pinching both sides of the tire and giving it a little twist, every six inches to a foot, all the way around the tire till you get to where the tire won’t slide into the rim.
Third, and this will really help, hold the rim parallel to the ground with both hands on either side. You want the section you’re struggling with farthest away from you. Now, place that section of rim down on the ground so the wheel is standing upright. Take the heel of both hands and starting at the top of the tire, slide your hands along the tire toward the ground – you’re effectively pushing any slack toward the section of tire you’re struggling with. Done with a little force, this will help a ton.
Finally, and this is the inside pool of the pro, it’s not about lifting the tire bead into the rim – you’re not strong enough (nor am I, that’s why I reached for the tire irons!). The key is to push the tube up so you don’t pinch it between the bead and the rim (this will likely result in wasting a CO2 canister as well as you’ve just pinched a hole into your precious tube… so now you’re double hit). Once you get the tube out of the way – you can give yourself a little slack by taking some of the bead out of the rim – then you roll the tire into its home. Grip with your fingers and roll with the pads of your hand. It should roll right in there with a little fuss and effort.
The keys are, again, confidence. You don’t need the tire irons. Center the tire and work any slack to where you need it. Clear the tube out of the way and roll the tire into place.