SENTENCE FRAGMENTS: this interaction sounds like robots greeting each other. a more natural flow could be
"Here Mitsuha-onee-sama, I have a letter from my father for you"
"Wow, an invitation to the royal palace!"


this whole bubble just feels awkward. im sure there's a better way to rewrite it but with the least changes i might go with
"There are many people lined up for this audience. Well, naturally there are others receiving rewards, but some of the great lords are just here for appearances."


make sure to keep your tenses consistent, these bubbles switch over from past to present tense at random.
"The Lord's armies near the invasion route who fought hard to stall the enemies' advance."
"scouts & messengers…"
"the decisive battle was in front of the main gate…"


another awkward sounding bubble. this might just have a missing word but could be rephrased to sound more natural
"actually there is one person who should be rewarded before me"
or
"actually there is someone more deserving of rewards than I"


and another awkward sounding speech with more missing words
"he threw himself in front of the bolt that was aimed at commander ice bringer and I"
"a courageous, loyal, young aristocrat"


proper tenses again, even though he is speaking about a past event, he should be speaking in the present tense
"is that so?"


grammar, in this case "dead" would be an adjective that you are attempting to use as a verb. also this sentence is rather awkward.
"yes, without that person's actions I would be dead now"
or
"yes, if they did not take action I would have died right there"


more missing words & jumbled sentence order.
"although he belongs to our kingdom's aristocratic order..."