TL;DR: Turn-based strategy games tend to be designed for high difficulty. As a less experienced gamer, how do I improve my gameplay in TBS games to meet these high standards?
I've been playing Fire Emblem Awakening for quite some time, and I've finished the game at Normal/Classic without excessive difficulty and without losing any units. I'm now playing at Hard/Casual, and I'm starting to have trouble—I'm losing units almost every battle, and while Casual mode limits the impact of these losses to the current battle, I feel like I'm doing things wrong. (Note that I am relying on pairing up considerably, I do check enemy movement and attack power to control casualties, and I do try to sequence moves so that attacks bring down enemy units as efficiently as possible. I have the EXPonential Growth and The Golden Gaffe DLC packs, but I'm trying to rely on them as little as possible.)
The Fire Emblem franchise has had a very high difficulty throughout most of its history. While recent titles have offered lower difficulty options, the high difficulty of many of these games has been well accepted, at least in Japan. A large portion of the Fire Emblem community, from what I can tell reading forum posts online, plays Awakening at difficulty levels higher than Normal, and Casual mode is strongly frowned upon.
This isn't limited to Fire Emblem. Other turn-based strategy games, like Battle For Wesnoth and XCOM: Enemy Unknown, tend to be designed for high difficulty (though to a lesser extent with Wesnoth).
This leads me to believe that my skill is not up to the standards historically set by TBS games and the community of players who play them.
So, in the spirit of this question, as a less experienced gamer, how do I improve my gameplay in turn-based strategy games? While an answer related to Fire Emblem Awakening is most relevant to me, I'd like to have tips that apply to TBS games in general.