While it is true that blood thinners can increase your risk of bleeding, it's also true that patients are carefully monitored for signs of excessive bleeding and usually you would undergo blood tests to ensure that you're not at high risk for bleeding. (I say "usually" because some of the newer blood thinners have a more predictable effect on blood clotting; consequently, it's often not necessary to do as much blood work.) You can help to lower your risk for bleeding by taking your medications only as prescribed, using a soft toothbrush, and, if you shave, using an electric shaver rather than a razor.
The sort of "holy grail" of anticoagulant (blood thinner) therapy would be to have a medication that's oral, reversible, and has a predictable effect on blood clotting so patients don't need bloodwork as frequently. The most that we have managed to achieve right now is 2 out of the 3. Here are some of the most common medications used:
Heparin - this is usually given by injection or IV. It achieves only one out of our three criteria - it's reversible. The antidote for too much heparin is protamine sulfate. Sometimes patients will be put on IV heparin initially when they have a confirmed or suspected DVT; this is because heparin is fairly fast-acting. Bloodwork will be done frequently and the IV rate will be adjusted as necessary. We call this a "heparin protocol" as there is a very specific algorithm that we follow to make adjustments.
Low molecular weight heparin (LMWH) - this is a group of medications that includes dalteparin (Fragmin), enoxaparin (Lovenox), and tinzaparin (Innohep). These are related to heparin and the antidote is the same, but they have a more predictable effect on blood clotting. They are given by injection, but usually once a day (heparin has to be given twice a day if it's given by injection). The reason that heparin and LMWH have to be injected is that heparin is a protein. If it's taken orally, the body just takes it apart and reassembles the amino acids into a different protein, rendering it useless. (The same thing is true of insulin, interestingly.)
Coumadin (warfarin) - this is actually the active ingredient in rat poison, but it's used very carefully! This is reversible and it's given orally, usually once a day. It's a very common practice to start a patient on IV heparin and also start them on Coumadin, because Coumadin takes 3 days to work. Unfortunately, the effect on blood clotting can be unpredictable, and doses can vary widely. (I have actually seen patients whose "standard" dose was to take one dose on certain days of the week and a different dose on other days!) It requires frequent bloodwork, and you have to be careful of how much vitamin K you take in because it's the antidote for too much Coumadin. This means some dietary restrictions, mostly for dark green vegetables such as broccoli as they are high in vitamin K.
Direct oral anticoagulants (DOACs) - these are getting closer to the "holy grail". They're relatively new and have only been around for a few years. These include apixaban, rivaroxaban, and dabigatran. They're oral and given once a day. They're not reversible, but the effect on clotting is fairly predictable, so hemorrhage is uncommon and patients don't usually need extensive bloodwork.
Hope this helps.