This is something that I do. Putting is an action word. It is a decisive thing I do. Putting is picking up something from one place and placing it somewhere else.
Putting my trust in the Lord is picking up my cares and concerns and placing them on Him and letting them rest there. It is picking up what is troubling me and putting it into His hands. It is the same as casting my cares upon Him. It is moving the cares onto Him. It is transferring all of my cares from one place (on me) to another (on Him).
1 Peter 5:6-7 – Therefore humble yourselves under the mighty hand of God, that He may exalt you in due time, casting all your care upon Him, for He cares for you.
Notice all of the scriptures that mention “Putting” your trust in the Lord.
Psalms 9:10 – And they who know Your name [who have experience and acquaintance with Your mercy] will lean on and confidently put their trust in You, for You, Lord, have not forsaken those who seek (inquire of and for) You.
Psalms 31:1 – In You, O Lord, do I put my trust and seek refuge; let me never be put to shame or [have my hope in You] disappointed; deliver me in Your righteousness!
Psalms 73:28 – But it is good for me to draw near to God; I have put my trust in the Lord God and made Him my refuge, that I may tell of all Your works.
John 4:50 – Jesus answered him, go in peace; your son will live! And the man put his trust in what Jesus said and started home.
Psalms 11:1 – In the Lord I put my trust; how can you say to my soul, flee as a bird to your mountain?
2 Chronicles 20:20 – They rose early in the morning and went out to the wilderness of Tekoa; and when they went out, Jehoshaphat stood and said, listen to me, O Judah and inhabitants of Jerusalem, put your trust in the Lord your God and you will be established. Put your trust in His prophets and succeed.
If you want more info on what this means or how to do it, see my article on How to Cast Your Cares.