Problem: The order of creation is contradictory

It is claimed that the order of events in Genesis 1 and 2 are different. Here is day-by-day the creation order in Genesis 1:

The plants problem

Genesis 2, however, might seem to place humans before plants or before animals. Here's Genesis 2:5-7:

When no bush of the field was yet in the land and no small plant of the field had yet sprung up - for the LORD God had not caused it to rain on the land, and there was no man to work the ground, and a mist was going up from the land and was watering the whole face of the ground - then the LORD God formed the man of dust from the ground and breathed into his nostrils the breath of life, and the man became a living creature. (ESV)

This seems to say that man was created before plants, but strictly speaking it only says that plants were not yet "in the land" or had not "sprung up". An anonymous reader of this page suggests that this could mean they existed but in some sort of dormant state, having not yet been "planted"; though this is hardly the most natural reading.

Another suggestion is that plants "of the field" refers specifically to cultivated plants. This argument has some weight, since the passage notes that there was nobody "to work the ground" - i.e. there were no farmers yet.

The animals problem

Then the LORD God said, 'It is not good that the man should be alone; I will make him a helper as his partner.' So out of the ground the LORD God formed every animal of the field and every bird of the air, and brought them to the man to see what he would call them; and whatever the man called each living creature, that was its name. (NRSV)

This appears to place the creation of man before animals, yet "the LORD God formed every animal" could simply mean "God created new specimens of creatures already in existence", though this is not the natural reading. Alternatively, the ESV and some other Bibles solve the problem by describing this creation in the past tense:

Then the LORD God said, "It is not good that the man should be alone; I will make him a helper fit for him." Now out of the ground the LORD God had formed every beast of the field and every bird of the heavens and brought them to the man to see what he would call them. And whatever the man called every living creature, that was its name. (ESV)

This way, God had already created these creatures. I'm not an expert in ancient Hebrew and can't say whether this translation is legitimate or not.

Conclusion

As you probably know if you're reading this, I generally argue against infallibilism. In this case I strongly suspect the two Genesis accounts had different authors with different understandings. However, I feel that this view is not really proven by a nuanced reading of the text.