Galatians 6:2 (KJV) Bear ye one another's burdens, and so fulfil the law of Christ.

Galatians 6:5 (KJV) For every man shall bear his own burden.

This is a situation where words do not always translate over their exact meaning from one language to another. Though we do not see it in English, Paul is employing two different Greek words for "burden." In Galatians 6:2 Paul uses the term, "baros" (meaning, "weight, or burden of the day" - see Matt. 20:12; Acts 15:28; 1 Thess. 2:7; Rev. 2:24) and in Galatians 6:5 he uses the term, phortion (meaning, "a ship's cargo" - see Acts 27:10). As stated in a translation handbook by Arichea and Nida:

The burden in verse 2 seems to refer to a load which is heavy and somewhat oppressive, and therefore it would be for the persons own good if he could get rid of it. In verse 5, however, Paul uses a word (load) which is used of a ships cargo or a mans pack or a soldiers knapsack. Accordingly, some see in this verse the figure of a soldier going to battle and bearing his own kit. The emphasis then would be on everyone doing his own share of normal duty.

As Christians we should assist others with oppressive loads, though I would not go as far as Arichea in saying it would be good to get rid of it. Loads are for our good (Rom. 8:28) and are bearable (1 Cor. 10:13), but assistance (Ecc 4:12; Rom. 12:15) in bearing them at times is needed, is helpful, and builds community. So, being strengthened while running the course would be a proper biblical perspective, as would being responsible for our own loads. As John Stott offers:

So we are to bear one another's burdens which are too heavy for a man to bear alone, but there is one burden which we cannot share - indeed do not need to because it is a pack light enough for every man to carry himself - and that is our responsibility to God on the day of Judgment. On that day you cannot carry my pack and I cannot carry yours.

References

Arichea, D. C., & Nida, E. A. A Handbook on Paul's Letter to the Galatians - UBS Handbook Series; Helps for Translators (149). New York: United Bible Societies, 1993.

John Stott. The Message of Galatians - Only One Way in Bible Speaks Today Series, pp. 159-160. Leicester UK/Downers Grove IL, Inter-Varsity Press, 1968.