Do native speakers use "relative clauses" much when speaking?
- "The bag he is carrying is very heavy."
- "Have you seen the photos Ann took?"
Do native speakers often talk like this?
Yes. See Frequency of Basic English Grammatical Structures: A Corpus Analysis (Roland et al. 2009):
In the Switchboard corpus, representing unscripted conversation, they found 25,440 relative clauses per million noun phrases.
In the Wall Street Journal corpus, representing formal written English, they found 46,788 relative clauses per million noun phrases.
So relative clauses are frequent in both informal conversation and formal written English, but they're more frequent in the latter.
Your examples are perfectly natural. All native speakers use relative clauses, and they're quite common in daily speech.
I have italicised the defining relative clauses:
"The bag (that) he is carrying is very heavy."
"Have you seen the photos (that) Ann took?"
They are perfectly natural. The version with that omitted is more informal.
Yes, these types of sentences are used daily in American English.
Slightly more common would be "He is carrying a very heavy bag." or "Have you seen Ann's photos?" However, the sentences have identical meanings and usages, with only a small difference in focus. One draws our attention to the subject ("He" and "You") and the other to the object ("the bag" and "the photos.")